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(a) the function IC is lower semicontinuous and proper convex; (b) if G ⊆ X is open, then

n 1 1 lim inf ln P Xi ∈ G ≥− inf {IC(w)}; n↑∞ n n w∈G " i=1 # X (c) if F ⊆ X is compact, open convex, or closed convex, then

1 1 n lim sup ln P Xi ∈ F ≤− inf {IC(w)}. n n w∈F n↑∞ " i=1 # X Furthermore, if X is finite-dimensional, then this bound is valid for any closed set F provided that E[eξkX1k] < +∞ for some number ξ > 0. Earlier, Donsker and Varadhan [6] proved Cram´er’s theorem under the stringent expo- nential moment condition E[eξkX1 k] < +∞ for all ξ > 0. Importantly, they showed that under this condition the upper bound in part (c) holds for any closed set F even when X is infinite-dimensional. Along with the use as stochastic processes, discrete-time renewal models find application in Equilibrium Statistical Physics with a different interpretation of the time coordinate. In particular, they are employed in studying the phenomenon of polymer pinning, whereby a polymer consisting of t ≥ 1 monomers is pinned by a substrate at the monomers T1,T2,... that represent renewed events along the polymer chain [7,8]. Supposing that the monomer Ti contributes an energy −v(Si) provided that Ti ≤ t, v being a real function over {1, 2,...} ∪ {∞} called the potential, the state of the polymer is described by the perturbed law Pt defined on the measurable space (Ω, F) by the Gibbs change of measure

dP eHt t := , dP Zt 1 where Ht := i≥1 v(Si) {Ti≤t} is the Hamiltonian and the normalizing constant Zt := E[eHt ] is the partition function. The model (Ω, F, P ) is called the pinning model (PM) and P t generalizes the original renewal model corresponding to the potential v = 0. The theory of large deviations we develop in this paper is framed within the PM supplied with the hypotheses of aperiodicity and extensivity. The waiting time distribution p := P[S1 = · ] is said to be aperiodic if P[S1 < ∞] > 0 and there is no proper sublattice of {1, 2,...} containing the support of p. We point out that a generic p can be made aperiodic when P[S1 < ∞] > 0 by simply changing the time unit. Assumption 1.1. The waiting time distribution p is aperiodic.

We say that the potential v is extensive if there exists a real number zo such that v(s) zos e p(s) ≤ e for all s. For instance, any potential v with the property that sups≥1{v(s)/s} < +∞ is extensive. Extensive potentials are the only that serve Equilibrium Statistical Physics, E Ht 1 v(t) where the partition function Zt ≥ [e {S1=t}] = e p(t) is expected to grow exponen- tially in t in order to define the free energy [7, 8]. Assumption 1.2. The potential v is extensive. Together with the PM we consider the constrained pinning model (CPM) where the last Pc monomer is always pinned by the substrate [7, 8]. It corresponds to the law t defined on the measurable space (Ω, F) through the change of measure

dPc U eHt t := t , P c d Zt

2 1 where Ut := i≥1 {Ti=t} is the renewal indicator, which takes value 1 if t is a renewal and value 0 otherwise, and Zc := E[U eHt ] is the partition function. Our interest in the CPM is P t t twofold. On the one hand, it turns out to be an effective mathematical tool to tackle the PM. Indeed, we can obtain a large deviation principle within the CPM by an argument based on convexity and super-additivity, and then transfer it to the PM by conditioning. The mentioned argument is a generalization of the approach to Cram´er’s theorem by Bahadur and Zabell [5], which in turn can be traced back to the method of Ruelle [9] and Lanford [10] for proving the existence of various thermodynamic limits. On the other hand, the CPM is a significant framework in itself because it is the mathematical skeleton of the Poland-Scheraga model of DNA denaturation and of some relevant lattice models of Statistical Mechanics, as discussed by the author in Ref. [11] where use of the theory developed in the present paper is made. These models are the cluster model of fluids proposed by Fisher and Felderhof, the model of protein folding introduced independently by Wako and Saitˆofirst and Mu˜noz and Eaton later, and the model of strained epitaxy considered by Tokar and Dreyss´e. The macroscopic observables that enter the thermodynamic description of these systems turn out to be cumulative rewards corresponding to rewards of the order of magnitude of the waiting times [11]. Before introducing our main results, we must say that the CPM is not well-defined a priori. In fact, it may happen with full probability that the time t is not a renewal, so that c c Zt = 0. However, assumption 1.1 resulting in Zt > 0 for every sufficiently large t settles the problem at least for all those t. To verify this fact, we observe that aperiodicity of p entails that there exist m coprime integers σ1,...,σm such that p(σl) > 0 for each l. The bound n n n c E Ht 1 v(si) c Zt ≥ [Ute i=1 {Si=si}] = i=1 e p(si) if t = i=1 si yields Zt > 0 whenever t is an integer conical combination of σ1,...,σm. On the other hand, the Frobenius number Q Q P tc ≥ 0 associated with σ1,...,σm is finite since these integers are coprime and by definition c any t>tc can be expressed as an integer conical combination of them. It follows that Zt > 0 for all t>tc.

1.2 Statement of main results This section reports the main results of the paper. In the sequel, assumptions 1.1 and 1.2 are tacitly supposed to be satisfied and the topological dual X ⋆ of X is understood as a Banach space with the norm induced by k·k. Let z be the function that maps each linear functional ϕ ∈ X ⋆ in the extended real number z(ϕ) defined by

R E ϕ(X1)+v(S1)−ζS1 1 z(ϕ) := inf ζ ∈ : e {S1<∞} ≤ 1 , (1.1)  h i  where the infimum over the empty set is customarily interpreted as +∞. The following proposition puts this function into context by relating z to the scaled cumulant generating function of Wt within the CPM. According to this proposition, z(0) turns out to be the free energy of the CPM [7,8] and, more in general, z(ϕ) can be regarded as the free energy of a ϕ(X1) CPM with the (possibly non-extensive) potential v + ln E[e |S1 = · ]. Proposition 1.1. The function z is proper convex and lower semicontinuous. The following limit holds for every ϕ ∈ X ⋆:

1 ϕ(Wt)+Ht lim ln E Ute = z(ϕ). t↑∞ t  Pc Ec Denoting the expectation with respect to the law t by t , proposition 1.1 entails that Ec ϕ(Wt) ⋆ limt↑∞(1/t) ln t [e ]= z(ϕ) − z(0) for all ϕ ∈ X , so that z − z(0) is exactly the scaled cumulant generating function of Wt within the CPM. We stress that the number z(0) is finite. E v(S1)−ζS1 1 P Indeed, we have [e {S1<∞}] > 1 for all sufficiently negative ζ as [S1 < ∞] > 0 E v(S1)−ζS1 1 v(s)−ζs by assumption 1.1 and, at the same time, [e {S1<∞}]= s≥1 e p(s) ≤ 1 for all ζ ≥ zo + ln 2, zo being the number introduced by assumption 1.2. The function z is finite everywhere in the following case, which is relevant for Statistical MePchanics as it comprises the macroscopic observables that enter the thermodynamic description of the system [11].

3 Example 1.1. The function z is finite everywhere if the reward X1 is dominated by the waiting time S1 in the sense that kX1k≤ MS1 with full probability for some constant M < ⋆ E ϕ(X1)+v(S1)−ζS1 1 +∞. This follows from the facts that, for any given ϕ ∈ X , [e {S1<∞}] ≥ E −MkϕkS1+v(S1)−ζS1 1 P [e {S1<∞}] > 1 for all sufficiently negative ζ, as [S1 < ∞] > 0 by E ϕ(X1)+v(S1)−ζS1 1 Mkϕks+v(s)−ζs assumption 1.1, and [e {S1<∞}] ≤ s≥1 e p(s) ≤ 1 for all ζ ≥ zo + Mkϕk + ln 2 with zo given by assumption 1.2. P We use the function z to construct a rate function. Let I be the Fenchel-Legendre transform of z − z(0), which associates every point w ∈ X with the extended real number I(w) given by I(w) := sup ϕ(w) − z(ϕ)+ z(0) . (1.2) ϕ∈X ⋆ n o The following theorem extends Cram´er’s theorem to the cumulative reward Wt with respect to the CPM and constitutes our first main result. It is proved together with proposition 1.1 in Section 2. Theorem 1.1. The following conclusions hold: (a) the function I is lower semicontinuous and proper convex; (b) if G ⊆ X is open, then

1 Pc Wt lim inf ln t ∈ G ≥− inf {I(w)}; t↑∞ t t w∈G   (c) if F ⊆ X is compact, open convex, closed convex, or any convex set in B(X ) when X is finite-dimensional, then

1 Pc Wt lim sup ln t ∈ F ≤− inf {I(w)}. t t w∈F t↑∞   Furthermore, if X is finite-dimensional, then this bound is valid for any closed set F E ξkX1k+v(S1)−ζS1 1 provided that [e {S1<∞}] < +∞ for some numbers ζ ≥ 0 and ξ > 0. The lower bound in part (b) and the upper bound in part (c) are called, respectively, large deviation lower bound and large deviation upper bound [12,13]. When a lower semicontinuous function I exists so that the large deviation lower bound holds for each open set G and the large deviation upper bound holds for each compact set F , then Wt is said to satisfy a weak large deviation principle (weak LDP) with rate function I [12, 13]. If the large deviation upper bound holds more generally for every closed set F , then Wt is said to satisfy a full large deviation principle (full LDP) [12,13]. Theorem 1.1 states that the cumulative reward Wt satisfies a weak LDP with rate function I given by (1.2) within the CPM. If in addition X E ξkX1k+v(S1)−ζS1 1 is finite-dimensional and the exponential moment condition [e {S1<∞}] < +∞ is fulfilled for some ζ ≥ 0 and ξ > 0, as certainly occurs in example 1.1 for any ξ > 0 and ζ >Mξ + zo, then Wt satisfies a full LDP. Regarding the validity of a full LDP for general infinite-dimensional Banach spaces X , finding sufficient conditions is a harder problem that will be the focus of future studies. Trying to sketch an analogy with Cram´er’s theorem and the work by Donsker and Varadhan [6], one should probably investigate situations where E ξkX1k+v(S1)−ζS1 1 there exists ζ ≥ 0 such that [e {S1<∞}] < +∞ for all ξ > 0. Let us move now to the PM, where there is no constraint on the last monomer. At variance with the CPM, the scaled cumulant generating function of Wt may not exist in the PM, but the following proposition, which is proved in Section 3, shows that at least some P P bounds hold true. Set ℓi := lim inf t↑∞(1/t) ln [S1 > t] and ℓs := lim supt↑∞(1/t) ln [S1 > t], and bear in mind that −∞ ≤ ℓi ≤ ℓs ≤ 0. Proposition 1.2. The following bounds hold for all ϕ ∈ X ⋆:

1 ϕ(W )+H z(ϕ) ∨ ℓi ≤ lim inf ln E e t t t↑∞ t

1  ϕ(W )+H  ≤ lim sup ln E e t t ≤ z(ϕ) ∨ ℓs. t↑∞ t   4 Denoting by Et the expectation with respect to Pt, proposition 1.2 entails that the limit ϕ(Wt) limt↑∞(1/t) ln Et[e ] exists, and equals z(ϕ) ∨ ℓs − z(0) ∨ ℓs, if either ℓi = ℓs or z(ϕ) ≥ ℓs. Thus, the scaled cumulant generating function of Wt with respect to the PM is defined if either ℓi = ℓs, which includes the case ℓs = −∞, or the condition z(ϕ) ≥ ℓs > −∞ is met for all ϕ ∈ X ⋆, as in the following example.

⋆ Example 1.2. The bound z(ϕ) ≥ ℓs > −∞ holds for all ϕ ∈ X if P[S1 < ∞]=1, lim infs↑∞ v(s)/s =0, and there exists a positive real function g on {1, 2,...}∪{∞} such that lims↑∞ g(s)/s = 0 and kX1k ≤ g(S1) with full probability. Indeed, given any ζ<ℓs, under these hypotheses one can find ǫ> 0 such that ζ + ǫ<ℓs ≤ 0 and −kϕkg(s)+ v(s) ≥−ǫs for E ϕ(X1)+v(S1)−ζS1 1 −kϕkg(s)+v(s)−ζs all sufficiently large s. Then, [e {S1<∞}] ≥ s≥1 e p(s) ≥ e−(ζ+ǫ)sp(s) ≥ e−(ζ+ǫ)tP[S > t] for all sufficiently large t as P[S = ∞]=0. It s>t 1 P 1 follows that E[eϕ(X1)+v(S1)−ζS1 1 ]=+∞ since ζ + ǫ<ℓs, which results in z(ϕ) ≥ ℓs P {S1<∞} according to definition (1.1). In order to establish large deviation bounds with respect to the PM, it is convenient to distinguish the case ℓs = −∞ from the case ℓs > −∞. The following theorem, which represents our second main result, provides weak and full LDPs for the renewal-reward process t 7→ Wt with respect to the PM when ℓs = −∞. The proof is given in Section 3.

Theorem 1.2. Assume ℓs = −∞. The following conclusions hold: (a) if G ⊆ X is open, then

1 Wt lim inf ln Pt ∈ G ≥− inf I(w) ; t↑∞ t t w∈G    (b) if F ⊆ X is compact, then

1 Wt lim sup ln Pt ∈ F ≤− inf I(w) . t t w∈F t↑∞    If F ⊆ X is open convex, closed convex, or any convex set in B(X ) when X is finite- dimensional, then this bound is valid whenever I(0) < +∞. Furthermore, if X is finite- E ξkX1k+v(S1)−ζS1 1 dimensional, then it is valid for any closed set F provided that [e {S1<∞}] < +∞ for some numbers ζ ≥ 0 and ξ > 0. In general, the large deviation upper bound in part (b) cannot be extended to convex sets if ℓs = −∞ and I(0) = +∞. Examples with an open convex set and a closed convex set where such bound fails will be shown at the end of Section 3. The case ℓs > −∞ is more involved and calls for two rate functions, Ii and Is, which are defined for each w ∈ X by the formulas

Ii(w) := sup ϕ(w) − z(ϕ) ∨ ℓi + z(0) ∨ ℓs (1.3) ϕ∈X ⋆ n o and Is(w) := sup ϕ(w) − z(ϕ) ∨ ℓs + z(0) ∨ ℓi . (1.4) ϕ∈X ⋆ n o The following theorem, which is our third and last main result, provides large deviation bounds with respect to the PM when ℓs > −∞. The proof is reported in Section 3.

Theorem 1.3. Assume ℓs > −∞. The following conclusions hold:

(a) the functions Ii and Is are lower semicontinuous and proper convex; (b) if G ⊆ X is open, then

1 Wt lim inf ln Pt ∈ G ≥− inf Ii(w) ; t↑∞ t t w∈G    5 (c) if F ⊆ X is compact, open convex, closed convex, or any convex set in B(X ) when X is finite-dimensional, then

1 Wt lim sup ln Pt ∈ F ≤− inf Is(w) . t t w∈F t↑∞    Furthermore, if X is finite-dimensional, then this bound is valid for any closed set F E ξkX1k+v(S1)−ζS1 1 provided that [e {S1<∞}] < +∞ for some numbers ζ ≥ 0 and ξ > 0.

Theorem 1.3 states that the renewal-reward process t 7→ Wt satisfies a weak LDP with rate function Is within the PM provided that Ii = Is. The exponential moment condition E ξkX1k+v(S1)−ζS1 1 [e {S1<∞}] < +∞ for some ζ ≥ 0 and ξ > 0 gives a full LDP with rate function Is when X is finite-dimensional and Ii = Is. We have Ii = Is if ℓi = ℓs, as expected ⋆ in most applications, or if the condition z(ϕ) ≥ ℓs is fulfilled for all ϕ ∈ X , as in example 1.2. In the latter case, Ii = Is = I.

1.3 Discussion Large deviations for renewal-reward processes have been investigated by many authors over the past decades. Their attention has been focused on both discrete-time and continuous- time frameworks and, in most cases, on rewards taking real values. In order to fix ideas, when talking about renewal systems in the domain of time we think of a PM with waiting times satisfying P[S1 < ∞] = 1 and potential v = 0. An almost omnipresent hypothesis in previous works is the Cram´er condition E[eξkX1k+ξS1 ] < +∞ for some number ξ > 0. The simplest example of renewal-reward process has unit rewards and corresponds to 1 the counting renewal process t 7→ Nt := i≥1 {Ti≤t}. Glynn and Whitt [14] investigated the connection between LDPs of the inverse processes t 7→ Nt and i 7→ Ti, providing a full P LDP for Nt under the Cram´er condition. This condition was later relaxed by Duffield and Whitt [18]. Jiang [20] studied the large deviations of the extended counting renewal process

1 α t 7→ i≥1 {Ti≤i t} with α ∈ [0, 1) under the Cram´er condition. Glynn and Whitt [14] and Duffield and Whitt [18], together with Puhalskii and Whitt [17], also investigated the P connection between sample-path LDPs of the processes t 7→ Nt and i 7→ Ti under the Cram´er condition. Starting from sample-path LDPs of inverse and compound processes, Duffy and Rodgers- Lee [19] sketched a full LDP for renewal-reward processes with real rewards by means of the contraction principle under the stringent exponential moment condition E[eξkX1k+ξS1 ] < +∞ for all ξ > 0. Some full LDPs for real renewal-reward processes were later proposed by Macci [15, 16] under existence and essentially smoothness of the scaled cumulant gener- ating function, which allow for an application of the G¨artner-Ellis theorem [12, 13]. Essen- tially smoothness of the scaled cumulant generating function has been recently relaxed by Borovkov and Mogulskii [21, 22], which used the Cram´er’s theorem to establish a full LDP under the Cram´er condition. Under the Cram´er condition, they [23–25] have also obtained sample-path LDPs for real renewal-reward processes. A different approach based on empirical measures has been considered by Lefevere, Mar- iani, and Zambotti [26], which have investigated large deviations for the empirical measures of forward and backward recurrence times associated with a renewal process, and have then derived by contraction a full LDP for renewal-rewards processes with rewards determined by the waiting times: Xi := f(Si) for each i with a bounded real function f. Later, Mariani and Zambotti [27] have developed a renewal version of Sanov’s theorem by studying the empirical law of rewards that take values in a generic Polish space. By appealing to the contraction principle, this result could give a full LDP for a renewal-reward process with rewards valued in a real separable Banach space, but only provided that the strong exponential moment condition E[eξkX1k] < +∞ for all ξ > 0 is satisfied as discussed by Schied [28]. We conclude this brief review of previous contributions by mentioning that a moderate deviation principle for real renewal-reward processes was obtained by Tsirelson [29] under an exponential moment condition. Exact asymptotics for the counting renewal process and real renewal-reward processes has been investigated under the Cram´er condition and several

6 additional smoothness hypotheses by Serfozo [30], Kuczek and Crank [31], Chi [32], and Borovkov and Mogulskii [33, 34]. Previous works leave open the question of whether some large deviation principles free from exponential moment conditions can be established for renewal-reward processes, in the wake of the sharp version of Cram´er’s theorem demonstrated by Bahadur and Zabell [5]. The present paper gives a positive answer to this question at the price of restricting to the discrete-time framework. Indeed, through theorems 1.1, 1.2, and 1.3 we supply weak LDPs and large deviation upper bounds for measurable convex sets that are completely free from hypotheses. Moreover, when finite-dimensional rewards are considered, and when P[S1 < ∞] = 1 and v = 0 to make a comparison with previous studies, we provide full LDPs under the exponential moment condition E[eξkX1k−ζS1 ] < +∞ for some numbers ζ ≥ 0 and ξ > 0, which is weaker than the Cram´er condition E[eξkX1k+ξS1 ] < +∞ for some ξ > 0. For instance, rewards of example 1.1 that define the macroscopic observables of Statistical Mechanics [11] always satisfy our weak exponential moment condition, whereas in general they do not fulfill the Cram´er condition. In order to drop exponential moment conditions, a novel approach with respect to past methods had to be devised to tackle the problem, and a new approach was suggested to us by the theory of polymer pinning [7,8]. This new approach is based on super-additivity, but requires discrete time to be implemented. It came from here the need to focus on the discrete- time framework. In such framework, conditioning on the event that the last time is a renewal time is a meaningful procedure and enables a super-additivity property of renewal-reward processes to emerge. This procedure introduces a constrained model similarly to what is done with polymers. This way, we were able to find a successful strategy for investigating large deviations and we were naturally led to link renewal-reward processes to the PM and the CPM. Importantly, the CPM is not a merely mathematical tool to tackle the PM, but it also represents the renewal models of Statistical Mechanics [11], such as the Poland- Scheraga model, the Fisher-Felderhof model, the Wako-Saitˆo-Mu˜noz-Eaton model, and the Tokar-Dreyss´emodel. In this respect, the large deviation theory developed in this paper must be added to those already existing for other models of Statistical Mechanics, including the Curie-Weiss model [35], the Curie-Weiss-Potts model [36], the mean-field Blume-Emery- Griffiths model [37], and, to some extent, the Ising model as well as general Gibbs measures relative to an interaction potential [38–41]. Going back for a moment to the domain of time with P[S1 < ∞] = 1 and v = 0, it is interesting to point out that Duffy and Rodgers-Lee [19], Lefevere, Mariani, and Zambotti [26], and Borovkov and Mogulskii [21, 22] found, with increasing level of generality, an apparently different rate function. They constructed the rate function for renewal-reward processes from the Cram´er rate function ΥC of the waiting time and reward pair (S1,X1), defined for each pair (β, w) ∈ R × X by

ϕ(X1)−ζS1 ΥC(β, w) := sup ϕ(w) − βζ − ln E e . (ζ,ϕ)∈R×X ⋆ n  o Starting from ΥC, they considered the function infγ>0{γΥC(·/γ, ·/γ)}, whose lower-semicontinuous regularization Υ is given for every (β, w) ∈ R × X by

Υ(β, w) := lim inf inf inf γΥC(α/γ,u/γ) . δ↓0 α∈(β−δ,β+δ) u∈Bw,δ γ>0  Here Bw,δ := {u ∈ X : ku − wk < δ} is the open ball of center w and radius δ. Duffy and Rodgers-Lee [19] dealt with the case X = R and ℓs = −∞ under a strong exponential moment condition, obtaining the rate function Λ := Υ(1, · ). In this case we have the rate ϕ(X1)−ζS1 function I by theorem 1.2 with z(ϕ) = inf{ζ ∈ R : E[e ] ≤ 1} for all ϕ as S1 < ∞ with full probability and v = 0. Lefevere, Mariani, and Zambotti [26] too found the rate function Λ when X1 = f(S1) with a bounded real function f. This instance falls under the umbrella of example 1.2, and so we get again the rate function I by theorem 1.3. Borovkov and Mogulskii [21, 22] studied the case X = R under the Cram´er condition. They obtained the rate function Λ when ℓs = −∞ and the rate function Λs := infβ∈[0,1]{Υ(β, ·) − (1 − β)ℓs} ⋆ when ℓi = ℓs > −∞ or z(ϕ) ≥ ℓs > −∞ for all ϕ ∈ X . In these cases we have the rate

7 functions I and Is, respectively, by theorem 1.2 and 1.3. Despite different expressions, our results are consistent with all the findings of these authors. Indeed, while uniqueness of the rate function [12, 13] suggests that there is at least some situation where I = Λ and Is = Λs, a direct comparison shows that these identities hold in general, as established by the following lemma which is proved in A.

Lemma 1.1. Assume that P[S1 < ∞]=1 and that v =0. Then, the following conclusions hold for every w ∈ X : (a) I(w) =Λ(w) := Υ(1, w);

(b) Is(w)=Λs(w) := infβ∈[0,1]{Υ(β, w) − (1 − β)ℓs} provided that ℓs > −∞. As a final remark, we stress that Borovkovand Mogulskii [21,22] opted for not introducing two different rate functions for the large deviation lower and upper bounds, thus considering only problems where Ii = Is. At variance with them, we decided to provide optimal large deviation bounds with possibly different rate functions in order to even address situations where the tail of the waiting time distribution is very oscillating. For instance, a physical renewal model giving rise to two possibly different rate functions has been found by Lefevere, Mariani, and Zambotti [42, 43] in the description of a free particle interacting with a heat bath.

2 Proof of proposition 1.1 and theorem 1.1

We prove proposition 1.1 and theorem 1.1 as follows. In Section 2.1 we show the existence of a weak LDP with a convex rate function. This is the step where convexity and super- additivity arguments come into play. In Section 2.2 we introduce the generalized renewal equation formalism, which allows us to express the scaled cumulative generating function in terms of the function z defined by (1.1). Then, we use this formalism in Section 2.3 to also relate the rate function to z. Finally, in Section 2.4 we summarize the results linking them to proposition 1.1 and to parts (a), (b), and (c) of theorem 1.1. Our theory of large deviations take advantage of the fact that a renewal process forgets t t the past and starts over at every renewal. Concretely, this means that (Uτ+t, ∆τ H, ∆τ W )t≥1 t t with ∆τ H := Hτ+t−Hτ and ∆τ W := Wτ+t−Wτ is independent of (Hτ , Wτ ) and distributed as (Ut,Ht, Wt)t≥1 conditional on the event that a given integer τ ≥ 1 is a renewal, namely

E 1 1 t t U = E 1 U {(Hτ ,Wτ )∈·} {(Uτ+t,∆τ H,∆τ W )t≥1∈⋆} τ {(Hτ ,Wτ )∈·} τ h i Eh1 i · {(Ut,Ht,Wt)t≥1∈⋆} . (2.1) h i A formal proof of (2.1) can be drawn by noticing that if τ = Tn for some positive in- teger n, then Ti ≤ τ for each i ≤ n and Ti > τ for any i>n. It follows that Hτ = n n i=1 v(Si) and Wτ = i=1 Xi, so that the random vector (Hτ , Wτ ) depends only on 1 (S1,X1),..., (Sn,Xn). At the same time, for any t ≥ 1 we have Uτ+t = {T =τ+t} = P P i≥n+1 i 1 , ∆t H = v(S )1 = v(S )1 , i≥1 {Sn+1+···+Sn+i=t} τ i≥n+1 i {Ti≤τ+t} i≥1 n+Pi {Sn+1+···+Sn+i≤t} and analogously ∆t W = X 1 = X 1 , showing P τ i≥n+1P i {Ti≤τ+t} i≥1 Pn+i {Sn+1+···+Sn+i≤t} that the vector (U , ∆t H, ∆t W ) depends only on (S ,X ), (S ,X ),... through τ+t τ P τ Pn+1 n+1 n+2 n+2 the same formula that connects (Ut,Ht, Wt) to (S1,X1), (S2,X2),....

2.1 Weak LDP in the constrained setting

c We leave the normalizing constant Zt aside for the moment and focus on the measure µt over B(X ) defined for each time t ≥ 1 by

E 1 Ht µt := W Ute . t ∈·  t  

8 E Ht c We have µt(X )= [Ute ]= Zt > 0 for all t>tc and some tc ≥ 0 thanks to assumption 1.1 about aperiodicity, as we have seen at the end of Section 1.1. Of fundamental importance is Pc c the following super-multiplicativity property, which is not fulfilled by t [Wt/t ∈ · ]= µt/Zt precisely because of normalization. Lemma 2.1. Let C ∈ B(X ) be convex and let τ ≥ 1 and t ≥ 1 be two integers. Then, µτ+t(C) ≥ µτ (C) · µt(C).

t Proof. Writing Wτ+t/(τ + t) = λWτ /τ + (1 − λ)∆τ W/t with λ := τ/(τ + t), we recognize t that Wτ+t/(τ + t) ∈ C whenever Wτ /τ ∈ C and ∆τ W/t ∈ C since C is convex. It follows that

E 1 Hτ+t µτ+t(C) = Wτ+t Uτ+te ∈C  τ+t  E 1 1 Hτ+t ≥ W ∆t W Uτ+te τ ∈C τ ∈C  τ t  t E 1 Hτ 1 ∆τ H = W e ∆t W Uτ+te . τ ∈C τ ∈C  τ t    A looser lower bound is obtained by introducing the renewal indicator Uτ with the motivation t t that (Uτ+t, ∆τ H, ∆τ W ) is independent of (Uτ ,Hτ , Wτ ) and distributed as (Ut,Ht, Wt) when τ is a renewal. This way, invoking (2.1) we find

t E 1 Hτ 1 ∆τ H µτ+t(C) ≥ W Uτ e ∆t W Uτ+te τ ∈C τ ∈C  τ t  E 1 Hτ E 1 Ht = W Uτ e · W Ute τ ∈C t ∈C  τ   t  = µτ (C) · µt(C ),  which proves the lemma. Super-multiplicativity, which becomes super-additivity once logarithms are taken, makes it possible to describe in general terms the exponential decay with t of the measure µt. To this purpose, we denote by L the extended real function over B(X ) defined by the formula

1 L := sup ln µ . t t t>tc  

If C ∈B(X ) is convex, then the super-additivity of ln µt(C) immediately gives lim supt↑∞(1/t) ln µt(C)= L(C). The following lemma improves this result when C is open as well as convex. Hereafter we denote by Bw,δ := {u ∈ X : ku − wk <δ} the open ball of center w and radius δ, which is an example of open convex set.

Lemma 2.2. Let C ⊆ X be open and convex. Then, limt↑∞(1/t) ln µt(C) exists as an extended real number and is equal to L(C). Proof. We shall show in a moment that the hypothesis that C is open entails that either µt(C) = 0 for all t>tc or there exists τ ≥ tc such that µt(C) > 0 for all t > τ. Lemma 2.2 is obvious in the first case. The second case is solved as follows. Pick an integer s>tc. Then, fix an integer γ ≥ 1 such that γs>τ and a constant M > −∞ such that ln µr(C) ≥ M when γs ≤ r< 2γs, which exists because γs>τ. Expressing any t ≥ 2γs as t = qγs+r with q ≥ 1 and γs ≤ r < 2γs, super-additivity gives ln µt(C) ≥ qγ ln µs(C) + ln µr(C) ≥ qγ ln µs(C)+ M, thus showing that lim inft↑∞(1/t) ln µt(C) ≥ (1/s) ln µs(C). The arbitrariness of s yields lim inft↑∞(1/t) ln µt(C) ≥ sups>tc {(1/s) ln µs(C)} =: L(C). We now prove that either µt(C) = 0 for all t>tc or there exists τ ≥ tc with the property that µt(C) > 0 for all t > τ. Assume that µτo (C) > 0 for some τo > tc. To begin with, we notice that if for every w ∈ C it were possible to find a number δw > 0 such that µτo (Bw,δw ) = 0, then the open covering {Bw,δw }w∈C of C would contain a countable

9 subcollection covering C by separability of X and Lindel¨of’s lemma with the consequence that µτo (C) = 0. This argument shows that there exists at least one point wo ∈ C such that µτo (Bwo,δ) > 0 for all δ > 0. Since C is open, there is δo > 0 such that Bwo,2δo ⊆ C.

This way, we have constructed open balls Bk := Bwo,kδo so that µτo (B1) > 0 and B2 ⊆ C. c Furthermore, since limk↑∞ µr(Bk)= µr(X ) = Zr > 0 for all r>tc, there exists an integer ko ≥ 1 such that µr(Bko ) > 0 if r satisfies τo ≤ r< 2τo. Set τ := 2koτo. Let us pick an arbitrary t > τ and let us show that µt(C) > 0. The fact that t > τ ≥ 2τo makes it possible to express t as t = qτo + r with integers q and r such that q ≥ 1 and r τo ≤ r < 2τo. We notice that Wt/t ∈ B2 whenever Wqτo /qτo ∈ B1 and ∆qτo W/r ∈ Bko , as the following bounds demonstrate:

r Wt − two ≤ Wqτo − qτowo + ∆qτo W − rwo < δo(qτo + kor) <δ o( t +2koτo)= δo (t + τ) < 2δot.

Then, recalling that B2 ⊆ C we get

E 1 Ht µt(C) ≥ Wt Ute ∈B2  t 

 Ht E 1 1 ∆r W ≥ Wqτo qτo Ute ∈B1 ∈Bk  qτo r o  r  Hqτ ∆ H E 1  o 1 ∆r W qτo = Wqτo e qτo Uqτo+r e . ∈B1 ∈Bk  qτo r o  As in the proof of lemma 2.2, a convenient looser lower bound is obtained by introducing r r Uqτo . Since (Uqτo+r, ∆qτo H, ∆qτo W ) is independent of (Uqτo ,Hqτo , Wqτo ) and distributed as (Ur,Hr, Wr) when qτo is a renewal we find

r Hqτ ∆ H E 1 o 1 ∆r W qτo µt(C) ≥ Wqτo Uqτo e qτo Uqτo+r e ∈B1 ∈Bk  qτo r o 

E 1 Hqτo E 1 Hr = Wqτo Uqτo e · Wr Ure ∈B1 ∈Bk  qτo   r o  = µ (B ) · µ (B ) ≥ µq (B ) · µ (B ), qτo  1 r ko τo 1 r ko where the last inequality is due to super-multiplicativity because B1 is convex. We deduce from here that µt(C) > 0 as both µτo (B1) > 0 and µr(Bko ) > 0 by construction. Lemma 2.2 suggests to consider the putative rate function J that maps any w ∈ X in the extended real number J(w) defined by

J(w) := − inf L(Bw,δ) . δ>0  In fact, the function J controls the measure decay of open and compact sets as follows. Proposition 2.1. The following conclusions hold: 1 (i) lim inf ln µt(G) ≥− inf {J(w)} for each G ⊆ X open; t↑∞ t w∈G 1 (ii) lim sup ln µt(K) ≤− inf {J(w)} for each K ⊆ X compact. t↑∞ t w∈K Proof. Part (i) is immediate. Let G ⊆ X be open, let w ∈ G be an arbitrary point, and let δ > 0 be such that Bw,δ ⊆ G. Since µt(G) ≥ µt(Bw,δ) and since Bw,δ is open and convex, lemma 2.2 gives lim inft↑∞(1/t) ln µt(G) ≥ limt↑∞(1/t) ln µt(Bw,δ) = L(Bw,δ) ≥ −J(w). The conclusion follows from the arbitrariness of w. Moving to part (ii), pick a compact set K in X and assume infw∈K{J(w)} > −∞, otherwise there is nothing to prove. Let λ < infw∈K {J(w)} be a real number. Since there exists ǫ > 0 such that λ + ǫ ≤ J(w) = − infδ>0{L(Bw,δ)} for every w ∈ K, a number

10 δw > 0 can be found for each w ∈ K in such a way that L(Bw,δw ) ≤ −λ. Then, lemma

2.2 yields limt↑∞(1/t) ln µt(Bw,δw ) ≤ −λ for such δw. Due to the compactness of K, there exist finitely many points w ,...,w in K such that K ⊆ ∪n B . It follows that 1 n i=1 wi,δwi µ (K) ≤ n µ (B ), which in turn gives lim sup (1/t) ln µ (K) ≤ −λ. This way, t i=1 t wi,δwi t↑∞ t we get the desired upper bound by sending λ to infw∈K {J(w)}. P The first important properties of J are presented in the following lemma. Lemma 2.3. The function J is lower semicontinuous and convex.

Proof. Pick w ∈ X and let {wi}i≥0 be a sequence of points converging to w. We show that lim infi↑∞ J(wi) ≥ −L(Bw,δ) for all numbers δ > 0, which results in lim inf i↑∞ J(wi) ≥ J(w) and proves the lower semicontinuity of J. Given δ > 0 there exists io ≥ 0 such that kwi − wk ≤ δ/2 if i ≥ io. Then, monotonicity of L inherited from the measures µt entails that −J(wi) ≤ L(Bwi,δ/2) ≤ L(Bw,δ) for each i ≥ io since Bwi,δ/2 ⊆ Bw,δ. The bound lim infi↑∞ J(wi) ≥ −L(Bw,δ) follows from here. As far as the proof of the convexity of J is concerned, lower semicontinuity combined with the fact that dyadic rationals in [0, 1] are dense in [0, 1] makes it sufficient to verify that for each u and w in X u + w J(u)+ J(w) J ≤ . (2.2) 2 2   To this aim, we notice that for each number δ > 0 and integer t ≥ 1 the conditions Wt/t ∈ t Bu,δ and ∆tW/t ∈ Bw,δ imply W2t/(2t) ∈ B(u+w)/2,δ, as one can easily verify. It follows that

H2t u+w E 1 µ2t B = W2t U2te 2 ,δ ∈B + 2t u w ,δ    2   H2t E 1 1 ∆t ≥ Wt tW U2te ∈Bu,δ ∈B  t t w,δ  t Ht ∆ H E 1  1 t t = Wt e ∆ W U2te . ∈Bu,δ t ∈B  t t w,δ   t  t Inserting Ut and exploiting the fact that (U2t, ∆tH, ∆tW ) is independent of (Ut,Ht, Wt) and distributed as (Ut,Ht, Wt) when t is a renewal we get

t E 1 Ht 1 ∆tH µ2t B u+w ≥ W Ute ∆tW U2te 2 ,δ t ∈B t t u,δ t ∈Bw,δ     E 1 Ht E 1 Ht = Wt Ute · Wt Ute ∈Bu,δ ∈Bw,δ  t   t  = µt(Bu,δ) · µt(B w,δ). 

This way, taking logarithms, dividing by 2t, and sending t to infinity, we find L(B(u+w)/2,δ) ≥ (1/2)L(Bu,δ)+(1/2)L(Bw,δ) ≥−(1/2)J(u) − (1/2)J(w) thanks to lemma 2.2 because open balls are open convex sets. Inequality (2.2) follows from here by the arbitrariness of δ. We conclude the section strengthening proposition 2.1 for convex sets. We know that lim supt↑∞(1/t) ln µt(C)= L(C) for every C ∈B(X ) convex thanks to super-additivity. The following lemma draws a link between L(C) and infw∈C{J(w)}. Lemma 2.4. Let C ⊆ X be open convex, closed convex, or any convex set in B(X ) when X is finite-dimensional. Then, L(C) ≤− infw∈C{J(w)}. Proof. The lemma is trivial if L(C) = −∞. Assume L(C) > −∞ and pick ǫ > 0. Since lim supt↑∞(1/t) ln µt(C) = L(C) there exists an integer τ ≥ 1 such that L(C) ≤ (1/τ) ln µτ (C)+ǫ. Completeness and separability of X entail that µτ is tight as it is bounded c from above by Zτ < +∞ (see [44], theorem 7.1.7). Consequently, a compact set Ko ⊆ C can be found so that µτ (C) ≤ µτ (Ko)+[1−exp(−ǫτ)]µτ (C). Thus, µτ (C) ≤ exp(ǫτ)µτ (Ko) and L(C) ≤ (1/τ) ln µτ (Ko)+2ǫ follows. We shall show in a moment that there exists a compact

11 convex set K with the property that Ko ⊆ K ⊆ C. Then, using the fact that Ko ⊆ K we reach the further bound L(C) ≤ (1/τ) ln µτ (K)+2ǫ ≤ L(K)+2ǫ. At this point, we notice that on the one hand L(K) = lim supt↑∞(1/t) ln µt(K) by super-additivity as K is convex, and on the other hand lim supt↑∞(1/t) ln µt(K) ≤ − infw∈K{J(w)} by part (ii) of propo- sition 2.1 as K is compact. Thus, L(C) ≤ − infw∈K{J(w)} +2ǫ ≤ − infw∈C{J(w)} +2ǫ because K ⊆ C and the lemma follows from the arbitrariness of ǫ. Let us prove now that there exists a compact convex set K with the property that Ko ⊆ K ⊆ C. The hypothesis that the convex set C is either open or closed when X if infinite- dimensional comes into play here. Let Co be the convex hull of Ko and let K := cl Co, cl A denoting the closure of a set A. Clearly, Ko ⊆ Co ⊆ C and Co ⊆ K. Since Ko is compact, Co is convex and compact whenever X is finite-dimensional, whereas K is convex and compact in any circumstance (see [45], theorem 3.20). We want to demonstrate that K ⊆ C. If X is finite-dimensional, then K = Co and we get the desired result from Co ⊆ C. If X is infinite- dimensional and C is closed, then K ⊆ C follows from Co ⊆ C by taking closures. The only nontrivial case is when X is infinite-dimensional and C is open. Assume that C is open from now on and for each w ∈ C let δw > 0 be such that Bw,2δw ⊆ C. As Ko is compact, there exist finitely many points w ,...,w in K so that K ⊆ ∪n B . Let K′ be the convex hull 1 n o o i=1 wi,δwi of ∪n (cl B ∩K). We have K′ ⊆ C because ∪n (cl B ∩K) ⊆ ∪n cl B ⊆ C i=1 wi,δwi i=1 wi,δwi i=1 wi,δwi thanks to the fact that B ⊆ C for every i and because C is convex. This way, wi,2δwi K ⊆ C is verified if we show that K = K′. The inclusion K′ ⊆ K is immediate since ∪n (cl B ∩ K) ⊆ K and K is convex. In order to show the opposite inclusion K ⊆ K′ i=1 wi,δwi we observe that the set K′ is convex and compact since it is the convex hull of the union of the compact convex sets cl Bw1,δw1 ∩K,..., cl Bwn,δwn ∩K (see [45], theorem 3.20). Then, we observe that K ⊆ K′ as ∪n (cl B ∩K) = (∪n cl B )∩K and both ∪n cl B o i=1 wi,δwi i=1 wi,δwi i=1 wi,δwi ′ and K contain Ko. This way, we first realize that Co ⊆ K since Co is the smallest convex set ′ ′ that contains Ko, and by taking closures we later deduce that K ⊆ K as K is closed.

2.2 Expectations and generalized renewal equation

Let (S1, V1), (S2, V2),... be a sequence of independent and identically distributed random vectors on (Ω, F, P), the Vi’s taking values in [0, +∞), and for each time t ≥ 1 denote by Ψt the expected value E 1 1 Ψt := Ut {Ti>t} + Vi {Ti≤t} . (2.3)  iY≥1   Here we determine the asymptotic exponential rate of growth of Ψt with respect to t. The solution to this problem is a needed preliminary step to relate the rate function J to the function z defined by (1.1). The computation of Ψt takes advantage of the generalized renewal equation t Ψt = asΨt−s (2.4) s=1 X E 1 satisfied for each t ≥ 1 with the initial condition Ψ0 := 1, where as := [V1 {S1=s}] is a non-negative extended real number. This equation is deduced conditioning on S1 and then using the fact that the renewal process starts over at the renewal time T1. We are only interested in the case where aσl > 0 for each l, σ1,...,σm being the m coprime integers introduced in Section 1.1 to make effective aperiodicity of the waiting time distribution. E −ζS1 1 −ζs The expected value A(ζ) := [V1e {S1<∞}] = s≥1 ase exists as an extended real number and defines a lower semicontinuous function A that maps ζ ∈ R in A(ζ). The number ψ given by P ψ := inf ζ ∈ R : A(ζ) ≤ 1 , (2.5) where the infimum over the empty setn is customarily interpretedo as +∞, exactly is the exponential rate of growth we are looking for as stated by the next proposition. The level m R −ζσl set {ζ ∈ : A(ζ) ≤ 1} is bounded from below since A(ζ) ≥ l=1 aσl e > 1 for all ζ sufficiently negative and closed due to lower semicontinuity. Consequently, ψ > −∞ and P

12 ψt A(ψ) ≤ 1 if ψ< +∞. It follows that Ψt ≤ e for all t ≥ 1, which is trivial if ψ =+∞ and is easily verified by induction starting from (2.4) when ψ< +∞.

Proposition 2.2. limt↑∞(1/t) ln Ψt exists as an extended real number and is equal to ψ > ψt −∞. Moreover, the bound Ψt ≤ e holds for all t ≥ 1.

ψt Proof. The bound Ψt ≤ e for all t ≥ 1 gives lim supt↑∞(1/t) ln Ψt ≤ ψ. Let us show that 1 lim inf ln Ψt ≥ ψ. (2.6) t↑∞ t E 1 1 n 1 n n We have Ψt ≥ [Ut i≥1( {Ti>t} + Vi {Ti≤t}) i=1 {Si=si}] = i=1 asi if t = i=1 si . c This way, the same arguments used in Section 1.1 to deduce Z > 0 for all t>tc yield Q Q tQ P Ψt > 0 for all t>tc as aσl > 0 by hypothesis for each l. This property allows us to prove (2.6) as follows. Pick a real number ζ < ψ and notice that there exists an integer τ ≥ 1 τ −ζs so that s=1 ase ≥ 1. On the contrary we would have A(ζ) ≤ 1, which contradicts the assumption that ζ < ψ. Since Ψt > 0 for all t>tc, we can find a constant M > −∞ such P that ln Ψt ≥ M + ζt for every t satisfying tc < t ≤ tc + τ. As a matter of fact, this bound is valid for all t>tc. Indeed, an argument by induction based on the generalized renewal equation (2.4) shows that if t>tc + τ and ln Ψt−s ≥ M + ζ(t − s) for any positive s ≤ τ, then t τ τ M+ζt −ζs M+ζt Ψt = asΨt−s ≥ asΨt−s ≥ e ase ≥ e . s=1 s=1 s=1 X X X It follows that lim inft↑∞(1/t) ln Ψt ≥ ζ, giving (2.6) once ζ is sent to ψ. The first application of proposition 2.2 we consider is concerned with the function z ⋆ ϕ(Xi)+v(Si) defined by (1.1). To this aim we pick a linear functional ϕ ∈ X and we set Vi := e E ϕ(X1)+v(S1)1 for every i. In this case, we have aσl = [e {S1=σl}] > 0 for each l as p(σl) > 0 ϕ(Wt)+Ht and Ψt = E[Ute ] for all t since

P 1 1 ϕ(Xi)+v(Si)1 i≥1[ϕ(Xi)+v(si)] {Ti≤t} ϕ(Wt)+Ht {Ti>t} + e {Ti≤t} = e = e . iY≥1 h i Moreover, a direct comparison with (1.1) shows that the number ψ associated with the present V1 by formula (2.5) is nothing but z(ϕ). Consequently, proposition 2.2 gives ϕ(Wt)+Ht limt↑∞(1/t) ln E[Ute ] = z(ϕ) and z(ϕ) > −∞. It follows from here thanks to the arbitrariness of ϕ that z is convex and that z never attains −∞, thus resulting in a proper convex function since z is finite at least in 0 due to assumption 1.2 as we have seen at ϕ(Wt)+Ht z(ϕ)t the beginning of Section 1.2. Proposition 2.2 also shows that E[Ute ] ≤ e for all t ≥ 1. The function z is lower semicontinuous because if {ϕi}i≥0 is a sequence converging z(ϕi)t ϕi(Wt)+Ht to ϕ and t is any positive integer, then the bound e ≥ E[Ute ] and Fatou’s ϕi(Wt)+Ht ϕ(Wt)+Ht lemma give lim infi↑∞ z(ϕi) ≥ lim infi↑∞(1/t) ln E[Ute ] ≥ (1/t) ln E[Ute ], which results in lim infi↑∞ z(ϕi) ≥ z(ϕ) when t is sent to infinity. We have thus proved the following lemma. Lemma 2.5. The function z is proper convex and lower semicontinuous. Given any ϕ ∈ X ⋆, ϕ(Wt)+Ht z(ϕ)t ϕ(Wt)+Ht the bound E[Ute ] ≤ e is valid for all t ≥ 1 and the limit limt↑∞(1/t) ln E[Ute ]= z(ϕ) holds.

2.3 Connection with the function z In this section we prove that the rate function J is the Fenchel-Legendre transform of z, namely that J(w) = supϕ∈X ⋆ {ϕ(w) − z(ϕ)} for all w ∈ X . Lemma 2.3 states that J is convex and lower semicontinuous. Actually, J is proper convex. Indeed, by combining lemma 2.2 with C := X and lemma 2.5 with ϕ := 0 we get L(X ) = limt↑∞(1/t) ln µt(X )= Ht limt↑∞(1/t) ln E[Ute ] = z(0). This way, part (i) of proposition 2.1 with G := X gives z(0) ≥ − infw∈X {J(w)} and lemma 2.4 with C := X yields z(0) ≤ − infw∈X {J(w)}, with

13 the consequence that infw∈X {J(w)} = −z(0). As z(0) is finite, this equality shows that J is finite at some point and that it never attains −∞. Proper convexity and lower semicontinuity allow us to express J in terms of its convex conjugate J ⋆ as follows (see [46], theorem 2.3.3):

J(w) = sup ϕ(w) − J ⋆(ϕ) (2.7) ϕ∈X ⋆ n o ⋆ ⋆ for every w ∈ X with J (ϕ) := supw∈X {ϕ(w) − J(w)} for all ϕ ∈ X . This way, in order to demonstrate that J is the Fenchel-Legendre transform of z it suffices to show that J ⋆ = z. Basically, this argument is the same argument used by Cerf and Petit [47] for a short proof of Cram´er’s theorem in R. Proving the bound J ⋆(ϕ) ≤ z(ϕ) for all ϕ ∈ X ⋆ is not difficult. To do this, we fix ϕ ∈ X ⋆ and we observe that lemma 2.5 together with the fact that ϕ(Wt − tw) ≥ −kWt − twkkϕk≥ −tδkϕk if Wt/t ∈ Bw,δ gives for every t ≥ 1, w ∈ X , and δ > 0

z(ϕ)t E ϕ(Wt)+Ht E 1 ϕ(Wt)+Ht e ≥ Ute ≥ Wt Ute t ∈Bw,δ h i   tϕ(w) E 1 ϕ(Wt−tw)+Ht = e Wt Ute ∈Bw,δ  t  tϕ(w)−tδkϕk E 1 Ht ≥ e Wt Ute ∈Bw,δ  t  tϕ(w)−tδkϕk = e µt(Bw,δ). Taking logarithms, dividing by t, and sending t to infinity, we get from here z(ϕ) ≥ ϕ(w)+ L(Bw,δ)−δkϕk≥ ϕ(w)−J(w)+δkϕk thanks to lemma 2.2. Thus, sending δ to zero first and appealing to the arbitrariness of w later we reach the bound z(ϕ) ≥ supw∈X {ϕ(w)−J(w)} =: J ⋆(ϕ). A more sophisticated use of proposition 2.2 leads to the opposite bound, and hence to equality as stated by the following proposition. Proposition 2.3. The convex conjugate J ⋆ of J equals z. Proof. Pick a linear functional ϕ ∈ X ⋆. As z(ϕ) ≥ J ⋆(ϕ), in order to show that z(ϕ)= J ⋆(ϕ) we must prove that z(ϕ) ≤ J ⋆(ϕ). Assume that J ⋆(ϕ) < +∞, otherwise there is nothing to prove. We are going to obtain the bound z(ϕ) ≤ J ⋆(ϕ) in two steps. At first we verify that for each K ⊆ X compact 1 E 1 ϕ(Wt)+Ht ⋆ lim sup ln Wt Ute ≤ J (ϕ). (2.8) t t ∈K t↑∞   Then, we demonstrate that for each real number ζJ ⋆(ϕ) and ρ > 0 be two real numbers. Since there exists ǫ> 0 such that ϕ(w) + infδ>0{L(Bw,δ)} = ϕ(w) − J(w) ≤ ⋆ J (ϕ) ≤ λ−ǫ for all w, for each w ∈ X we can find δw > 0 in such a way that δwkϕk <ρ and

L(Bw,δw ) ≤ λ − ϕ(w). Lemma 2.2 gives limt↑∞(1/t) ln µt(Bw,δw ) ≤ λ − ϕ(w) for such δw.

Furthermore, we have ϕ(Wt −tw) ≤kWt −twkkϕk≤ tδwkϕk < tρ if Wt/t ∈ Bw,δw . From the compactness of K there exist finitely many points w ,...,w in K so that K ⊆ ∪n B . 1 n i=1 wi,δwi It follows that for all t ≥ 1 n E 1 ϕ(Wt)+Ht E 1 ϕ(Wt)+Ht Wt Ute ≤ Wt Ute t ∈K t ∈Bwi,δw i=1 i   X    n  tϕ(wi) E 1 ϕ(Wt−twi)+Ht = e Wt Ute t ∈Bwi,δw i=1 i X   n  ≤ µ (B )etϕ(wi)+tρ. t wi,δwi i=1 X

14 By combining this bound with lim (1/t) ln µ (B ) ≤ λ − ϕ(w ) for each i we find t↑∞ t wi,δwi i 1 E 1 ϕ(Wt)+Ht lim sup ln Wt Ute ≤ λ + ρ. t t ∈K t↑∞   This way, we reach (2.8) by sending λto J ⋆(ϕ ) and ρ to 0. E ϕ(X1)+v(S1)−ζS1 1 We now verify (2.9). Pick a real number ζ E ϕ(X1)+v(S1)1 1 by definition of z(ϕ). Recall that [e {S1=σl}] > 0 for all l since the m coprime integers σ1,...,σm satisfy p(σl) > 0 for every l. We shall show at the end that there exists a compact convex set K ⊆ X such that

E ϕ(X1)+v(S1)−ζS1 1 1 e {X1/S1∈K} {S1<∞} > 1 (2.10) h i and E ϕ(X1)+v(S1)1 1 e {X1/S1∈K} {S1=σl} > 0 (2.11)

h ϕ(Xi)+v(Si)1 i for each l. This way, setting Vi := e {Xi/Si∈K} for all i and introducing the number ψ defined by

R E −ηS1 1 ψ := inf η ∈ : V1e {S1<∞} ≤ 1 ,  h i  E 1 we have ζ < ψ from (2.10). At the same time, if as := [V1 {S1=s}] for all s, then (2.11) gives aσl > 0 for each l. Consequently, we can invoke proposition 2.2 with the present Vi to get

1 E 1 1 ζ < ψ = lim ln Ut {T >t} + Vi {T ≤t} t↑∞ t i i  iY≥1   1 E ϕ(Wt)+Ht 1 1 1 = lim ln Ute {T >t} + {X /S ∈K} {T ≤t} . t↑∞ t i i i i  iY≥1   On the other hand, as K is convex, the condition Xi/Si ∈ K for all i such that Ti ≤ t entails Wt/t ∈ K when t is a renewal. To understand this point, we write Wt/t = 1 i≥1(Xi/Si)(Si/t) {Ti≤t} and we notice that when there exists a positive integer n such that T = t, then (S /t)1 = n (S /t)= T /t = 1. It follows that P n i≥1 i {Ti≤t} i=1 i n P1 P E ϕ(Wt)+Ht 1 1 1 ζ < lim ln Ute {T >t} + {X /S ∈K} {T ≤t} t↑∞ t i i i i  iY≥1   1 E ϕ(Wt)+Ht 1 1 1 1 = lim ln Ute Wt {Ti>t} + {Xi/Si∈K} {Ti≤t} t↑∞ t t ∈K  iY≥1   1  E ϕ(Wt)+Ht 1 ≤ lim sup ln Ute Wt , t t ∈K t↑∞   which proves (2.9).  To conclude the proof of the proposition, we must show the validity of (2.10) and (2.11) for some compact convex set K. To this aim, consider the finite measure πR := E ϕ(X1)+v(S1)−ζS1 1 1 1 [e {X1/S1∈·} {kX1k≤R} {S1≤R}] on B(X ), R being a positive real num- E ϕ(X1)+v(S1)−ζS1 1 ber. The fact that [e {S1<∞}] > 1 implies that there exists a sufficiently large R so that πR(X ) > 1 and completeness and separability of X entail that πR is tight (see [44], theorem 7.1.7). It follows that there exists a compact set Ko such that E ϕ(X1)+v(S1)−ζS1 1 1 πR(Ko) > 1, which gives [e {X1/S1∈Ko} {S1<∞}] ≥ πR(Ko) > 1. Simi- E ϕ(X1)+v(S1)1 1 1 lar arguments with πR := [e {X1/S1∈·} {kX1k≤R} {S1=σl}] in combination with E ϕ(X1)+v(S1)1 E ϕ(X1)+v(S1)1 1 [e {S1=σl}] > 0 yield [e {X1/S1∈Kl} {S1=σl}] > 0 for some compact set Kl. Let K be the closed convex hull of Ko ∪ K1 ∪···∪ Km. The set K is convex and compact (see [45], theorem 3.20) and satisfies (2.10) and (2.11) as Ko ⊆ K and Kl ⊆ K for each l.

15 2.4 Proposition 1.1 and theorem 1.1 point by point In this section we explicitly verify proposition 1.1 and theorem 1.1 point by point, but the former simply is part of lemma 2.5 and does not need other demonstrations. Lemma 2.3 states that J is convex and lower semicontinuous. Moreover, we have seen that J is proper convex at the beginning of the last section. As J(w) = supϕ∈X ⋆ {ϕ(w) − z(ϕ)} for all w ∈ X thanks to (2.7) and proposition 2.3, the rate function I defined by (1.2) equals J + z(0) and inherits the lower semicontinuity and proper convexity of J. These facts prove part (a) of theorem 1.1. Part (b) of theorem 1.1 follows from part (i) of propo- P c sitions 2.1 bearing in mind that ln t[Wt/t ∈ · ] = ln µt − ln Zt for each t>tc, that c E Ht limt↑∞(1/t) ln Zt = limt↑∞ [Ute ] = z(0) by lemma 2.5, and that I = J + z(0). Sim- ilarly, part (c) of theorem 1.1 concerning compact sets is due to part (ii) of proposition 2.1. Part (c) regarding convex sets follows from the limit lim supt↑∞(1/t) ln µt(C) = L(C) valid for any C ∈ B(X ) convex and lemma 2.4. Finally, part (c) for closed sets in the finite-dimensional case is demonstrated by the following proposition. Let us observe that E ξkX1k+v(S1)−ζS1 1 the exponential moment condition [e {S1<∞}] < +∞ for some numbers ζ ≥ 0 and ξ > 0 implies z(ϕ) < +∞ for all ϕ ∈ X ⋆ such that kϕk ≤ ξ. Indeed, the validity of this condition with certain ζ ≥ 0 and ξ > 0 entails that a number h −h E ξkX1k+v(S1)−ζS1 1 large enough can be found so that e [e {S1<∞}] ≤ 1. It follows that E ϕ(X1)+v(S1)−(ζ+h)S1 1 −h E ξkX1k+v(S1)−ζS1 1 [e {S1<∞}] ≤ e [e {S1<∞}] ≤ 1 if kϕk≤ ξ as S1 ≥ 1 with full probability, which gives z(ϕ) < +∞ according to definition (1.1). It is easy to verify that if X is finite-dimensional, then the above exponential moment condition is tantamount to the existence of ξ > 0 such that z(ϕ) < +∞ for all ϕ fulfilling kϕk≤ ξ. Proposition 2.4. Assume that X has finite dimension and that there exist numbers ζ ≥ 0 E ξkX1k+v(S1)−ζS1 1 and ξ > 0 such that [e {S1<∞}] < +∞. Then, for each F ⊆ X closed 1 lim sup ln µt(F ) ≤− inf {J(w)}. t↑∞ t w∈F

Proof. Fix a closed set F in X and observe that infw∈F {J(w)}≥ infw∈X {J(w)} = −z(0) > −∞. Then, pick a real number λ < infw∈F {J(w)}. Let d be the dimension of X , let ⋆ {w1,...,wd} be a basis of X , and let {ϑ1,...,ϑd} ⊂ X be the dual basis: ϑi(wj ) equals 1 if i = j and 0 otherwise for all i and j. For i ranging from 1 to d, set ϕi := ϑi/kϑik and ϕd+i := −ϕi. Since z(ϕ) < +∞ if kϕk ≤ ξ, ξ > 0 being the number associated with the hypothesized exponential moment condition, there exists a real number ρ > 0 with the property that z(ξϕi) − ξρ ≤ −λ for each i. Denoting by K the compact set 2d c 2d K := ∩i=1{w ∈ X : ϕi(w) ≤ ρ}, we have K = ∪i=1{w ∈ X : ϕi(w) > ρ}. This way, by ξϕi(Wt)+Ht z(ξϕi)t making use of the Chernoff bound first and of the bound E[Ute ] ≤ e due to lemma 2.5 later, we obtain

2d 2d c E 1 Ht µt(K ) ≤ µt w ∈ X : ϕi(w) >ρ = {ϕi(Wt)>ρt}Ute i=1 i=1 X   X h i 2d  d ξϕi(Wt)−ξρt+Ht z(ξϕi)t−ξρt −λt ≤ E Ute ≤ e ≤ 2de , i=1 i=1 X h i X c −λt giving µt(F )= µt(F ∩K)+µt(F ∩K ) ≤ µt(F ∩K)+2de for each t. On the other hand, part (ii) of proposition 2.1 with the compact set F ∩ K shows that lim supt↑∞(1/t) ln µt(F ∩ K) ≤− infw∈F ∩K{J(w)}≤− infw∈F {J(w)}≤−λ. It follows that lim supt↑∞(1/t) ln µt(F ) ≤ −λ, which proves the proposition once λ is sent to infw∈F {J(w)}.

3 Proof of proposition 1.2 and of theorems 1.2 and 1.3

Large deviation bounds within the PM can be made a consequence of the corresponding bounds in the CPM by exploiting conditioning as follows. Pick an integer time t ≥ 1 and notice that if T1 ≤ t, then there is one and only one positive integer n ≤ t such that Tn ≤ t

16 t and Tn+1 >t. Thus,Ω = {T1 >t}∪{T1 ≤ t} and {T1 ≤ t} = ∪n=1{Tn ≤ t and Tn+1 >t} = t t ∪n=1 ∪τ=n {Tn = τ and Tn+1 > t}, the events {Tn = τ and Tn+1 > t} for 1 ≤ n ≤ τ ≤ t being disjoint. The condition T1 > t is tantamount to S1 > t and implies that Ht = 0 and Wt = 0. The condition Tn = τ and Tn+1 > t is tantamount to Tn = τ and Sn+1 > t − τ n n and implies that Ht = i=1 v(Si)= Hτ and Wt = i=1 Xi = Wτ are independent of Sn+1. This way, for every ϕ ∈ X ⋆ we find the identity between measures P P

E 1 ϕ(Wt)+Ht E 1 1 ϕ(Wt)+Ht {Wt∈·}e = {Wt∈·} {S1>t}e h i th t i E 1 1 1 ϕ(Wt)+Ht + {Wt∈·} {Tn=τ} {Sn+1>t−τ}e n=1 τ=n h i 1X X P = {0∈·} · [S1 >t] t τ E 1 1 ϕ(Wτ )+Hτ P + {Wτ ∈·} {Tn=τ}e · [S1 >t − τ] τ=1 n=1 h i 1X X P = {0∈·} · [S1 >t] t E 1 ϕ(Wτ )+Hτ P + {Wτ ∈·}Uτ e · [S1 >t − τ]. (3.1) τ=1 X h i Formula (3.1) connects the free setting with the constrained setting and is the starting point to prove proposition 1.2 and theorems 1.2 and 1.3. Once again, we leave normalization aside at the beginning and focus on the measure νt := ZtPt[Wt/t ∈ · ] on B(X ). Identity (3.1) with ϕ = 0 results in the expression

ν = E 1 eHt t Wt ∈·  t   t 1 P E 1 Hτ P = {0∈·} · [S1 >t]+ Wτ Uτ e · [S1 >t − τ]. (3.2) t ∈· τ=1 X    We use this expression to derive a lower large deviation bound in Section 3.1 and an upper large deviation bound in Section 3.2. Theorem 1.2 is verified point by point in Section 3.3, where two counterexamples are also shown to demonstrate that the upper large deviation bound for open convex sets and closed convex sets cannot hold in general when ℓs = −∞ and I(0) = +∞. Finally, theorem 1.3 is verified point by point in Section 3.4. Regarding proposition 1.2, it is an immediate consequence of formula (3.1), which entails E ϕ(Wt)+Ht P t E ϕ(Wτ )+Hτ P [e ]= [S1 > t]+ τ=1 [Uτ e ] · [S1 >t − τ] for each t and ϕ. Indeed, P P recalling the definitions lim inft↑∞(1/t) ln [S1 >t] =: ℓi and lim supt↑∞(1/t) ln [S1 >t] =: P ϕ(Wt)+Ht ⋆ ℓs, as well as the limit limt↑∞(1/t) ln E[Ute ]= z(ϕ) due to lemma 2.5, for all ϕ ∈ X we get

1 ϕ(Wt)+Ht lim inf ln E e ≥ z(ϕ) ∨ ℓi (3.3) t↑∞ t and   1 ϕ(Wt)+Ht lim sup ln E e ≤ z(ϕ) ∨ ℓs. (3.4) t↑∞ t   3.1 The lower large deviation bound

In this section we prove the following lower bound without restrictions on ℓi and ℓs. Proposition 3.1. For each G ⊆ X open 1 lim inf ln νt(G) ≥− inf sup ϕ(w) − z(ϕ) ∨ ℓi . t↑∞ t w∈G ϕ∈X ⋆ n o

17 Proof. Pick an open set G in X . In order to demonstrate the proposition it suffices to verify that for all w ∈ G 1 lim inf ln νt(G) ≥− sup ϕ(w) − z(ϕ) ∨ ℓi . (3.5) t↑∞ t ϕ∈X ⋆ n o This bound is immediate when ℓi = −∞. Indeed, keeping only the term corresponding to τ = t in the r.h.s. of (3.2) we get νt(G) ≥ µt(G), which shows that lim inft↑∞(1/t) ln νt(G) ≥ −J(w) for any w ∈ G thanks to part (i) of proposition 2.1. On the other hand, J(w) is the r.h.s. of (3.5) if ℓi = −∞ by formula (2.7) and proposition 2.3. The proof of (3.5) is more laborious when ℓi > −∞ and we assume that ℓi > −∞ from now on. Let dom z := {ϕ ∈ X ⋆ : z(ϕ) < +∞} be the effective domain of z and consider the function F that for a given w ∈ G maps (β, ϕ) ∈ [0, 1]×dom z in the real number F (β, ϕ) := ϕ(w)−βz(ϕ)−(1−β)ℓi. The function F is concave and upper semicontinuous with respect to ϕ for each fixed β ∈ [0, 1], inheriting these properties from z, and convex and continuous with respect to β for each fixed ϕ ∈ dom z. Then, due to compactness of the closed interval [0, 1], Sion’s minimax theorem allows us to exchange the infimum over β ∈ [0, 1] and the supre- mum over ϕ ∈ dom z: supϕ∈dom z infβ∈[0,1] F (β, ϕ) = infβ∈[0,1] supϕ∈dom z F (β, ϕ) . As infβ∈[0,1]{ϕ(w) − βz(ϕ) − (1 − β)ℓi} = ϕ(w) − z(ϕ) ∨ ℓi and z(ϕ) ∨ ℓi =+∞ when ϕ∈ / dom z, this identity can be written as  

sup ϕ(w) − z(ϕ) ∨ ℓi = inf sup ϕ(w) − βz(ϕ) − (1 − β)ℓi . ϕ∈X ⋆ β∈[0,1] ϕ∈dom z n o n o This way, we get the bound (3.5) if we prove that for every w ∈ G and β ∈ [0, 1] 1 lim inf ln νt(G) ≥− sup ϕ(w) − βz(ϕ) − (1 − β)ℓi . (3.6) t↑∞ t ϕ∈dom z n o We prove (3.6) considering the case β > 0 first. Pick a point w ∈ G and a number β ∈ (0, 1] and denote by τt the greatest integer that is less than or equal to βt. Let δ > 0 be such that Bw,2δ ⊆ G and focus on all those sufficiently large integers t such that τt > 0 and kwk < βδt. Within this setting, we have that the event Wτt /τt ∈ Bw/β,δ implies Wτt /t ∈ Bw,2δ ⊆ G. Indeed, since 0 ≤ t − τt/β < 1/β and kwk < βδt we find kWτt − twk≤kWτt − (τt/β)wk + (t − τt/β)kwk < kWτt − (τt/β)wk + δt. It follows that if kWτt − (τt/β)wk < δτt, then kWτt − twk < δτt + δt ≤ 2δt. This way, keeping only the term corresponding to τ = τt > 0 in the r.h.s. of (3.2), we obtain

E 1 Ht P νt(G) ≥ Wτ Uτ e · S1 >t − τt t ∈G t  t    E 1 Ht P ≥ Wτt Uτt e · S1 >t − τt ∈Bw/β,δ  τt    = µ B w · P S >t − τ . (3.7) τt β ,δ 1 t     We have limt↑∞(1/τt) ln µτt (Bw/β,δ)= L(Bw/β,δ) ≥−J(w/β) by lemma 2.2. We also have limt↑∞ τt/t = β and lim inft↑∞(1/t) ln P[S1 >t − τt]=(1 − β)ℓi. The latter limit is trivial in the case β = 1 to which τt = t corresponds, whereas it follows from lim inft↑∞(1/t) ln P[S1 > t] =: ℓi when β < 1 due to the fact that t − τt is now diverging as t is sent to infinity. These arguments in combination with (3.7) prove that 1 lim inf ln νt(G) ≥ −βJ(w/β)+(1 − β)ℓi t↑∞ t

= − sup ϕ(w) − βz(ϕ) + (1 − β)ℓi ϕ∈X ⋆ n o = − sup ϕ(w) − βz(ϕ) − (1 − β)ℓi , ϕ∈dom z n o which is (3.6) under the hypothesis that β > 0.

18 In order to settle the case β = 0, we take a point u ∈ X such that c := J(u) is finite, ⋆ which exists because J is proper convex. We have z(ϕ)= J (ϕ) := supw∈X {ϕ(w)−J(w)}≥ ϕ(u) − c for all ϕ ∈ X ⋆ by proposition 2.3. As G is open, for a given w ∈ G we can find a number δ ∈ (0, 1) such that w + ǫu ∈ G whenever ǫ ∈ (0,δ). Then, the bound (3.6) applies with a positive ǫ<δ< 1 in place of β and w + ǫu in place of w to give 1 lim inf ln νt(G) ≥ − sup ϕ(w + ǫu) − ǫz(ϕ) − (1 − ǫ)ℓi t↑∞ t ϕ∈dom z n o ≥ − sup ϕ(w) − ℓi − ǫ(c + ℓi). ϕ∈dom z  We obtain (3.6) corresponding to β = 0 from here by sending ǫ to zero.

3.2 The upper large deviation bound An upper large deviation bound for compact sets can be proved by means of standard arguments from large deviation theory without distinguishing the case ℓs > −∞ from the case ℓs = −∞. The following result holds. Proposition 3.2. For each compact set K ⊆ X 1 lim sup ln νt(K) ≤− inf sup ϕ(w) − z(ϕ) ∨ ℓs . t↑∞ t w∈K ϕ∈X ⋆ n o Proof. Let K be a compact set in X and notice that infw∈K supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}≥ −z(0) ∨ ℓs > −∞. Let λ< infw∈K supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs} and ρ> 0 be real numbers. As there exists ǫ > 0 such that supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}≥ λ + ǫ for all w ∈ K, a linear ⋆ functional ϕw ∈ X can be found for each w ∈ K with the property that ϕw(w)−z(ϕw)∨ℓs ≥ λ. It is manifest that z(ϕw) < +∞ for such ϕw. Let δw > 0 be a number that satisfies

δwkϕwk≤ ρ. Then, for every positive integers t and τ ≤ t the condition Wτ /t ∈ Bw,δw entails ϕw(Wτ − tw) ≥ −kWτ − twkkϕwk > −δwkϕwkt ≥−ρt, namely ϕw(Wτ ) − tϕw(w)+ ρt ≥ 0. ϕw(Wτ )+Hτ z(ϕw)τ This way, bearing in mind that E[Uτ e ] ≤ e by lemma 2.5 we get for each w ∈ K and integers t and τ ≤ t

E 1 Hτ E ϕw(Wτ )−tϕw(w)+tρ+Hτ Wτ Uτ e ≤ Uτ e t ∈Bw,δw   h i  ≤ ez(ϕw)τ−tϕw(w)+tρ ≤ eτ[z(ϕw)∨ℓs]−tϕw(w)+tρ. (3.8) We also have for each w ∈ K and t 1 ≤ e−tϕw(w)+tρ (3.9) {0∈Bw,δw } because if 0 ∈ Bw,δw , then kwk <δw so that ϕw(w) ≤ δwkϕwk≤ ρ. Due to the compactness of K, there exist finitely many points w1,...,wn in K such that K ⊆ ∪n B . The facts that lim sup (1/t) ln P[S > t] =: ℓ and z(ϕ ) ∨ ℓ > −∞ i=1 wi,δwi t↑∞ 1 s wi s for each i ensure the existence of a positive constant M < +∞ such that for all t and i ≤ n

t[z(ϕw )∨ℓs]+tρ P[S1 >t] ≤ Me i . (3.10) At this point, identity (3.2) combined with (3.8), (3.9), and (3.10) shows that for every t n 1 P νt(K) ≤ {0∈Bw ,δ } · [S1 >t] i wi i=1 X n t E 1 Hτ P + Wτ Uτ e · S1 >t − τ t ∈Bwi,δw i=1 τ=1  i  X X   n t  τ[z(ϕ )∨ℓ ]−tϕ (w )+tρ (t−τ)[z(ϕ )∨ℓ ]+(t−τ)ρ ≤ M e wi s wi i · e wi s i=1 τ=0 X X n t t[z(ϕ )∨ℓ ]−tϕ (w )+2tρ −tλ+2tρ ≤ M e wi s wi i ≤ Mn(t + 1)e , i=1 τ=0 X X

19 which in turn yields lim supt↑∞(1/t) ln νt(K) ≤−λ +2ρ. The proposition follows from here by sending ρ to zero and λ to infw∈K supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}. The upper bound stated by proposition 3.2 cannot be extended in general to convex sets when ℓs = −∞. However, at least the following weaker upper bound holds for them. Lemma 3.1. Let C ⊆ X be open convex, closed convex, or any convex set in B(X ) when X is finite-dimensional. Then, for each real number ℓ ≥ ℓs 1 lim sup ln νt(C) ≤− inf sup ϕ(w) − z(ϕ) ∨ ℓ . t↑∞ t w∈C ϕ∈X ⋆ n o

Proof. Pick a real number ℓ ≥ ℓs and notice that infw∈C supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓ} ≥ −z(0) ∨ ℓ> −∞. Fix a real number λ< infw∈C supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓ}. To begin with, we observe that for any given real number η ≥ 1 and integer τ ≥ 1 we have the bound

ln µτ (ηC) ≤−λητ − ℓ(η − 1)τ, (3.11) where ηC := {ηw : w ∈ C}∈B(X ), which is convex, open if C is open, and closed if C is closed. Indeed, as there exists ǫ > 0 such that λ + ǫ ≤ supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓ} for all ⋆ w ∈ C, for every w ∈ C we can find ϕw ∈ X satisfying λ ≤ ϕw(w) − z(ϕw) ∨ ℓ. This way, for each w ∈ C we obtain

J(ηw) = sup ϕ(ηw) − z(ϕ) ≥ ηϕw(w) − z(ϕw) ϕ∈X ⋆  ≥ ηλ + η z(ϕw) ∨ ℓ − z(ϕw) ≥ ηλ + (η − 1)ℓ.   On the other hand, if γ is a large enough integer so that γτ >tc, then the convexity of ηC al- lows us to invoke super-additive properties to obtain (1/τ) ln µτ (ηC) ≤ (1/γτ) ln µγτ (ηC) ≤ L(ηC). Consequently, lemma 2.4 with the set ηC entails ln µτ (ηC) ≤ −τ infv∈ηC {J(v)}, which proves (3.11) because infv∈ηC {J(v)} = infw∈C{J(ηw)}≥ λη + ℓ(η − 1). We use the bound (3.11) as follows. Given any positive integers t and τ ≤ t, setting η := t/τ we have that Wτ /τ ∈ ηC is tantamount to Wτ /t ∈ C. This way, (3.11) yields

E 1 Hτ −λt−ℓ(t−τ) W Uτ e = µτ (ηC) ≤ e . (3.12) τ ∈C  t  For each t we also find  1 −λt−ℓt {0∈C} ≤ e (3.13) because if 0 ∈ C, then λ< supϕ∈X ⋆ {ϕ(w)−z(ϕ)∨ℓ} with w = 0 gives λ ≤ supϕ∈X ⋆ {−z(ϕ)∨ P ℓ}≤−ℓ. Finally, recalling that lim supt↑∞(1/t) ln [S1 >t] =: ℓs ≤ ℓ we realize that for any (ℓ+ρ)t fixed number ρ> 0 there exists a positive constant M < +∞ such that P[S1 >t] ≤ Me for all t ≥ 0. By making use of this bound in (3.2) as well as bounds (3.12) and (3.13) we find t −λt−ℓ(t−τ) (ℓ+ρ)(t−τ) −λt+ρt νt(C) ≤ M e · e ≤ M(t + 1)e . τ=0 X Thus lim supt↑∞(1/t) ln νt(C) ≤−λ+ρ, which proves the lemma once λ is sent to infw∈C supϕ∈X ⋆ {ϕ(w)− z(ϕ) ∨ ℓ} and ρ is sent to zero. We conclude the section demonstrating an upper large deviation bound for closed sets under the hypothesis that X is finite-dimensional and that an exponential moment condition holds. No restriction on ℓs is needed here. Proposition 3.3. Assume that X has finite dimension and that there exist numbers ζ ≥ 0 E ξkX1k+v(S1)−ζS1 1 and ξ > 0 such that [e {S1<∞}] < +∞. Then, for each F ⊆ X closed 1 lim sup ln νt(F ) ≤− inf sup ϕ(w) − z(ϕ) ∨ ℓs . t↑∞ t w∈F ϕ∈X ⋆ n o 20 Proof. Fix a closed set F in X and observe that infw∈F supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs} ≥ −z(0) ∨ ℓs > −∞. Pick a real number λ < infw∈F supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}. Let d be the dimension of X and let ϕ1,...,ϕ2d be the linear functionals introduced in the proof of proposition 2.4. Since z(ϕ) < +∞ if kϕk ≤ ξ by hypothesis, as we have seen in Section 2.4, there exists a positive number M < +∞ with the property that z(ξϕi) ≤ M for each i. Pick a number ρ > 0 such that M − ξρ ≤ −λ. Denoting by K the compact set 2d c 2d ∩i=1{w ∈ X : ϕi(w) ≤ ρ} we have K = ∪i=1{w ∈ X : ϕi(w) >ρ}. This way, starting from (3.2) and noticing that 0 ∈{/ w ∈ X : ϕi(w) >ρ} for all i, by using the Chernoff bound first ξϕ(Wt)+Ht z(ξϕi)t and the bound E[Ute ] ≤ e due to lemma 2.5 later we obtain

2d c νt(K ) ≤ νt w ∈ X : ϕi(w) >ρ i=1 X   2d t  E 1 Hτ P = {ϕi(Wτ )>ρt}Uτ e · [S1 >t − τ] i=1 τ=1 X X h i 2d t 2d t ξϕi(Wτ )−ξρt+Hτ z(ξϕi)τ−ξρt ≤ E Uτ e ≤ e i=1 τ=1 i=1 τ=1 X X h i X X ≤ 2dteMt−ξρt ≤ 2dte−λt,

c −λt which gives νt(F )= νt(F ∩ K)+ νt(F ∩ K ) ≤ νt(F ∩ K)+2dte for each t. On the other hand, proposition 3.2 with the compact set F ∩ K shows that lim supt↑∞(1/t) ln νt(F ∩ K) ≤ − infw∈F ∩K supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}≤− infw∈F supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs} ≤ −λ. Thus, lim supt↑∞(1/t) ln νt(F ) ≤ −λ and the proposition is proved by sending λ to infw∈F supϕ∈X ⋆ {ϕ(w) − z(ϕ) ∨ ℓs}.

3.3 Theorem 1.2 point by point and counterexamples

Now we explicitly verify theorem 1.2 point by point. Assume ℓs = −∞. Then, ℓi = −∞ Ht and starting from the facts that ln Pt[Wt/t ∈ · ] = ln νt − ln Zt and Zt := E[e ] for all t ≥ 1 we get part (a) of theorem 1.2 thanks to proposition 3.1 and formula (3.4) with ϕ = 0. Similarly, part (b) of theorem 1.2 for compact and closed sets is obtained by combining propositions 3.2 and 3.3 with formula (3.3). As far as convex sets is concerned, we observe that z(0) − I(0) = − supϕ∈X ⋆ {−z(ϕ)} = infϕ∈X ⋆ {z(ϕ)} so that z(ϕ) ≥ z(0) − I(0) for all ϕ ∈ X ⋆. This way, part (b) of theorem 1.2 for convex sets follows when I(0) < +∞ by invoking lemma 3.1 with ℓ := z(0) − I(0) and, again, formula (3.3) with ϕ = 0. The upper large deviation bound for convex sets cannot hold in general when ℓs = −∞ and I(0) = +∞. We show two examples where it fails, involving an open convex set and a closed convex set, respectively. In these examples we assume P[1 < S1 < ∞] = 1 and v = 0, so that Ht = 0, Zt = 1, and Pt[Wt/t ∈ · ] = νt for every t. The following is the counterexample with an open convex set.

Example 3.1. Consider the reward Xi := Si for each i. In this example we have X = R, so that for any ϕ ∈ X ⋆ there exists one and only one real number k such that ϕ(w)= kw for all w. As P[S1 < ∞]=1 and v =0, by identifying ϕ with k definitions (1.1) and (1.2) give z(k) = inf{ζ ∈ R : E[ekS1−ζS1 ] ≤ 1} = k for all k ∈ R, I(1) = 0, and I(w)=+∞ for each R 1 w ∈ \{1}. The rate function I is consistent with the fact that i≥1 Si {Ti≤t} = t if a re- newal occurs at time t. The upper bound lim sup (1/t) ln Pt[Wt/t ∈ C] ≤− infw∈C {I(w)} t↑∞ P does not hold with the open convex set C := (−∞, 1), for which infw∈C{I(w)} = +∞. In- deed, keeping only the term corresponding to τ = t − 1 in the r.h.s. of (3.2), observing that Wt−1/t =1 − 1/t ∈ C if Ut−1 =1, and recalling that P[S1 > 1]=1 by assumption, we find for each t ≥ 2 W 1 ≥ P t ∈ C = ν (C) t t t   E 1 Ht−1 P E Ht−1 ≥ Wt−1 Ut−1e · [S1 > 1] = Ut−1e , ∈C  t     21 giving limt↑∞(1/t) ln Pt[Wt/t ∈ C]=0 by lemma 2.5 as z(0) = 0. The following is the counterexample with a closed convex set.

Example 3.2. Consider the reward Xi := (Si, Yi) for each i with Yi independent of Si and distributed according to the standard Cauchy law: P[Yi ≤ y] = (1/π)[π/2 + arctan(y)] for all y ∈ R. In this example X = R2, so that for any ϕ ∈ X ⋆ there exists one and only one pair of real numbers k = (kS , kY ) such that ϕ(w) = kS wS + kY wY for all w = (wS , wY ). As P[S1 < ∞]=1 and v =0, and as Y1 has no exponential moments, by identifying ϕ with kS S1−ζS1 kY Y1 k definition (1.1) gives z(k) = inf{ζ ∈ R : E[e ] · E[e ] ≤ 1} = kS if kY = 0 and z(k)=+∞ if kY 6= 0. It follows from definition (1.2) that I(w)=0 if wS = 1 and P I(w)=+∞ otherwise. The upper bound lim supt↑∞(1/t) ln t[Wt/t ∈ C] ≤− infw∈C {I(w)} 2 does not hold with the closed convex set C := {w ∈ R : wS < 1 and wY ≥ 1/(1 − wS)}, for which infw∈C {I(w)} =+∞. Indeed, as we shall show in a moment we have for every t ≥ 2

Wt 1 ≥ P ∈ C = ν (C) ≥ P Y ≥ t2 · E U eHt−1 , (3.14) t t t 1 t−1       giving limt↑∞(1/t) ln Pt[Wt/t ∈ C]=0 by lemma 2.5 as z(0) = 0. In order to prove (3.14) we pick an integer t ≥ 2 and observe that when a renewal 1 occurs at the time t − 1, so that i≥1 Si {Ti≤t−1} = t − 1, then Wt−1/t ∈ C if and only if Y 1 ≥ t2. This way, keeping only the term corresponding to τ = t − 1 in the i≥1 i {Ti≤t−1} P r.h.s. of (3.2) and recalling that P[S1 > 1]=1 we get P

E 1 Ht−1 P νt(C) ≥ Wt−1 Ut−1e · [S1 > 1] ∈C  t  E 1 Ht−1 = 1 2 Ut−1e P Yi {T ≤t−1}≥t  i≥1 i  t−1  E 1 1 Ht−1 = n 2 {Tn=t−1}e Pi=1 Yi≥t n=1 X   t−1 n P 2 E 1 Ht−1 = Yi ≥ t · {Tn=t−1}e . n=1 " i=1 # X X h i n On the other hand, (1/n) i=1 Yi is distributed as Y1 by the stability property of the Cauchy law so that P t−1 n P 2 E 1 Ht−1 νt(C) ≥ Yi ≥ t · {Tn=t−1}e n=1 " i=1 # X X h i t−1 P 2 E 1 Ht−1 = nY1 ≥ t · {Tn=t−1}e n=1 X h i h i t−1 P 2 E 1 Ht−1 ≥ Y1 ≥ t · {Tn=t−1}e n=1 X h i h i 2 Ht−1 = P Y1 ≥ t · E Ut−1e .     3.4 Theorem 1.3 point by point

To conclude, we explicitly verify theorem 1.3 point by point. Assume ℓs > −∞. The functions Ii and Is defined by (1.3) and (1.4) are the Fenchel-Legendre transform of z ∨ ℓi − z(0) ∨ ℓs and z ∨ ℓs − z(0) ∨ ℓi, respectively. Convexity and lower semicontinuity of Ii and Is are immediate to check. The functions Ii and Is are proper convex. Indeed, considering for instance Ii, we have on the one hand Ii(w) ≥−z(0) ∨ ℓi + z(0) ∨ ℓs > −∞ for all w ∈ X , and on the other hand Ii(u) ≤ J(u)+z(0)∨ℓs < +∞ at some point u because J is proper convex. These arguments demonstrate part (a) of theorem 1.3. As far as part (b) and part (c) is

22 Ht concerned, we recall that ln Pt[Wt/t ∈ · ] = ln νt − ln Zt and that Zt := E[e ] for all t in such a way that part (b) follows from proposition 3.1 and formula (3.4) with ϕ = 0. Part (c) for compact and closed sets is due to propositions 3.2 and 3.3 combined with formula (3.3). Finally, part (c) for convex sets follows from lemma 3.1 with ℓ = ℓs and, again, formula (3.3).

Acknowledgements

The author is grateful to Paolo Tilli for useful discussions about the counterexamples pre- sented in Section 3.3 and to Francesco Caravenna and Paolo Dai Pra for valuable overall comments.

A Proof of lemma 1.1

Since S1 < ∞ with full probability and v = 0, according to definition (1.1) we have z(ϕ)= inf{ζ ∈ R : E[eϕ(X1)−ζS1 ] ≤ 1} for all ϕ ∈ X ⋆. We shall show that for every β ≥ 0 and w ∈ X Υ(β, w) = sup ϕ(w) − βz(ϕ) , (A.1) ϕ∈dom z n o where dom z := {ϕ ∈ X ⋆ : z(ϕ) < +∞} is the effective domain of the function z. The identity I = Λ immediately follows from (A.1) by taking β = 1 and proves part (a) of the lemma. Regarding part (b), assume ℓs > −∞ and consider the function F that for a given w ∈ X maps (β, ϕ) ∈ [0, 1] × dom z in the real number F (β, ϕ) := ϕ(w) − βz(ϕ) − (1 − β)ℓs. The function F is concave and upper semicontinuous with respect to ϕ for each fixed β ∈ [0, 1], inheriting these properties from z, and convex and continuous with respect to β for each fixed ϕ ∈ dom z. Then, due to compactness of the closed interval [0, 1], Sion’s minimax theorem allows us to exchange the infimum over β ∈ [0, 1] and the supremum over ϕ ∈ dom z: supϕ∈dom z infβ∈[0,1] F (β, ϕ) = infβ∈[0,1] supϕ∈dom z F (β, ϕ) . Since z(0) = 0 and z(ϕ)=+∞ if ϕ 6∈ dom z, this identity yields  

Is(w) = sup ϕ(w) − z(ϕ) ∨ ℓs = sup inf F (β, ϕ) ϕ∈X ⋆ ϕ∈dom z β∈[0,1] n o  = inf sup F (β, ϕ) = inf sup ϕ(w) − βz(ϕ) − (1 − β)ℓs β∈[0,1] ϕ∈dom z β∈[0,1] ϕ∈dom z  n o = inf Υ(β, w) − (1 − β)ℓs =: Λs(w). β∈[0,1] n o This way, part (b) of the lemma is demonstrated as w is an arbitrary point. Let us prove (A.1). Pick β ≥ 0 and w ∈ X . To begin with, we point out that the function that associates ζ with E[eϕ(X1)−ζS1 ] for a given ϕ ∈ X ⋆ is lower semicontinuous by Fatou’s lemma, so that E[eϕ(X1)−z(ϕ)S1 ] ≤ 1 if z(ϕ) < +∞. It follows that for any ϕ ∈ dom z

ϑ(X1)−ζS1 ΥC(β, w) := sup ϑ(w) − βζ − ln E e (ζ,ϑ)∈R×X ⋆ n  o ≥ ϕ(w) − βz(ϕ) − ln E eϕ(X1)−z(ϕ)S1 ≥ ϕ(w) − βz(ϕ), so that infγ>0{γΥC(β/γ,w/γ)} ≥ ϕ(w) − βz(ϕ). Continuity of ϕ results in Υ(β, w) ≥ ϕ(w) − βz(ϕ) and the arbitrariness of ϕ gives

Υ(β, w) ≥ sup ϕ(w) − βz(ϕ) . ϕ∈dom z n o The opposite bound, which leads us to the proof of (A.1), is more involved and is achieved through the following two inequalities: Υ(β, w) ≤ inf sup ϕ(w) − βζ − γ ln E eϕ(X1)−ζS1 (A.2) γ∈[0,β] (ζ,ϕ)∈D n  o ≤ sup ϕ(w) − βz(ϕ) , (A.3) ϕ∈dom z n o 23 where D := {(ζ, ϕ) ∈ R × X ⋆ : E[eϕ(X1)−ζS1 ] < +∞}. To get at (A.2) we observe that the definition of Υ immediately gives

Υ(β, w) ≤ sup ϕ(w) − βζ − γ ln E eϕ(X1)−ζS1 (A.4) (ζ,ϕ)∈D n  o for all γ > 0. The lower-semicontinuous regularization procedure used to construct Υ entails that this bound also holds for γ = 0, as we now show, thus giving (A.2). The rate function ΥC is proper convex by Cram´er’s theorem, so that there exist βo ∈ R and wo ∈ X such that ϕ(X1)−ζS1 ΥC(βo, wo) is finite. It follows that ϕ(wo) − βoζ − ln E[e ] ≤ ΥC(βo, wo) =: c ≥ 0, ϕ(X1)−ζS1 ⋆ that is ln E[e ] ≥ ϕ(wo) − βoζ − c for all ζ ∈ R and ϕ ∈ X . Then, for every δ > 0 and γo ∈ (0,δ) such that γo|βo| <δ and γokwok <δ we find

inf inf inf γΥC(α/γ,u/γ) ≤ inf γΥC(β/γ + γoβo/γ,w/γ + γowo/γ) α∈(β−δ,β+δ) u∈Bw,δ γ>0 γ>0

 ≤ γoΥC(β/γo + βo,w/γo + wo)

ϕ(X1)−ζS1 = sup ϕ(w + γowo) − (β + γoβo)ζ − γo ln E e (ϕ,ζ)∈D n  o ≤ sup ϕ(w) − βζ + γoc (ϕ,ζ)∈D n o ≤ sup ϕ(w) − βζ + δc. (ϕ,ζ)∈D n o It follows that

Υ(β, w) := lim inf inf inf γΥC(α/γ,u/γ) δ↓0 α∈(β−δ,β+δ) u∈Bw,δ γ>0  ≤ sup ϕ(w) − βζ , (ϕ,ζ)∈D n o which exactly is (A.4) when γ = 0. Let us move to bound (A.3), which we prove by invoking Sion’s minimax theorem once again. The function that associates (ζ, ϕ) ∈ D with E[eϕ(X1)−ζS1 ] is lower semicontinuous by Fatou’s lemma and convex, so that the real function that maps (γ,ζ,ϕ) ∈ [0,β] × D in ϕ(w) − βζ − γ ln E[eϕ(X1)−ζS1 ] is concave and upper semicontinuous with respect to (ζ, ϕ) for each fixed γ ∈ [0,β] and convex and continuous with respect to γ for each fixed pair (ζ, ϕ) ∈ D. Then, Sion’s theorem ensures us that

inf sup ϕ(w) − βζ − γ ln E eϕ(X1)−ζS1 γ∈[0,β] (ζ,ϕ)∈D n  o = sup inf ϕ(w) − βζ − γ ln E eϕ(X1)−ζS1 (ζ,ϕ)∈D γ∈[0,β] n  o = sup ϕ(w) − βζ − β ln 1 ∨ E eϕ(X1)−ζS1 . (ζ,ϕ)∈D n  o On the other hand, if (ζ, ϕ) ∈ D, then ϕ ∈ dom z because E[eϕ(X1)−(ζ+h)S1 ] ≤ e−hE[eϕ(X1)−ζS1 ] ≤ 1 for all sufficiently large h as S1 ≥ 1 with full probability. It follows that

inf sup ϕ(w) − βζ − γ ln E eϕ(X1)−ζS1 γ∈[0,β] (ζ,ϕ)∈D n  o = sup ϕ(w) − βζ − β ln 1 ∨ E eϕ(X1)−ζS1 (ζ,ϕ)∈D n  o ≤ sup sup ϕ(w) − βζ − β ln 1 ∨ E eϕ(X1)−ζS1 ϕ∈dom z ζ∈R n  o ≤ sup ϕ(w) − βz(ϕ) , ϕ∈dom z n o where the last bound demonstrates (A.3) and is justified as follows. Pick any ϕ ∈ dom z and recall that E[eϕ(X1)−ζS1 ] ≤ 1 or E[eϕ(X1)−ζS1 ] > 1 depending on whether ζ ≥ z(ϕ) or

24 ζ

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