Limit Theorems for the Fractional Non-Homogeneous Poisson Process

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∞ zn E (z)= , z C, α [0, 1). α Γ(αn + 1) ∈ ∈ n=0 X The Mittag-Leffler distribution was first considered in Gnedenko and Kovalenko (1968) and Khintchine (1969). A comprehensive treatment of the FHPP as a renewal process can be found in Mainardi et al. (2004) and Politi et al. (2011). Starting from the standard Poisson process N(t) as a point process, the FHPP can also be defined as N(t) time-changed by the inverse α-stable subordinator. Meerschaert et al. (2011) showed that both the renewal and the time- change approach yield the same stochastic process (in the sense that both processes have the same finite-dimensional distribution). Laskin (2003) and Beghin and Orsingher (2009, 2010) derived the governing equations associated with the one-dimensional distribution of the FHPP. In Leonenko et al. (2017), we introduced the fractional non-homogeneous Pois- son process (FNPP) as a generalization of the FHPP. The non-homogeneous Poisson process is an additive process with deterministic, time dependent intensity function and thus generally does not allow a representation as a classical renewal process. However, following the construction in Gergely and Yezhow (1973, 1975) we can define the FNPP as a general renewal process. Following the tine-change approach, the FNPP is defined as a non-homogeneous Poisson process time-changed by the inverse α-stable subordinator. Among other results, we have discussed in our previous work that the FHPP can be seen as a Cox process. Following up on this observation, in this article, we will show that, more generally, the FNPP can be treated as a Cox process dis- cussing the required choice of filtration. Cox processes or doubly stochastic processes (Cox (1955), Kingman (1964)) are relevant for various applications such as filtering theory (Brémaud, 1981), credit risk theory (Bielecki and Rutkowski, 2002) or actuarial risk theory (Grandell, 1991) and, in particular, ruin theory (Biard and Saussereau, 2014, 2016). Subsequently, we are able to identify the compensator of the FNPP. A similar generalization of the original Watanabe characterization (Watanabe, 1964) of the Poisson process can be found in case of the FHPP in Aletti et al. (2017). Limit theorems for Cox processes have been studied by Grandell (1976) and Serfozo (1972a,b). Specifically for the FHPP, scaling limits have been derived in Meerschaert and Scheffler (2004) and discussed in the context of parameter estimation in Cahoy et al. (2010).

The rest of the article is structured as follows: In Section 2 we give a short overview of deﬁnitions and notation concerning the fractional Poisson process. Section 3 and 4 are devoted to the application of the Cox process theory to the fractional Poisson process which allows us to identify its compensator and thus derive limit theorems via martingale methods. A diﬀerent approach to deriving asymptotics is followed in Section 5 and requires a regular variation condition imposed on the rate function of the fractional Poisson process. The fractional compound process is discussed in Section 6 where we derive both a one-dimensional limit theorem using Anscombe’s theorem and a functional limit. Finally, we give a brief discussion of simulation methods of the FHPP and verify the results of some of our results in a Monte Carlo experiment.

2 2. The fractional Poisson process

This section serves as a brief revision of the fractional Poisson process both in the homogeneous and the non-homogeneous case as well as a setup of notation.

Let (N1(t))t≥0 be a standard Poisson process with parameter 1. Deﬁne the function t Λ(s,t) := λ(τ) dτ, Zs where s,t 0 and λ : [0, ) (0, ) is locally integrable. For shorthand Λ(t) := Λ(0≥,t) and we assume∞ Λ(−→t) ∞for t . We get a non-homogeneous → ∞ → ∞ Poisson process (N(t))t≥0, by a time-transformation of the homogeneous Pois- son process with Λ: N(t) := N1(Λ(t)).

The α-stable subordinator is a Lévy process (Lα(t))t≥0 defined via the Laplace transform E[exp( uL (t))] = exp( tuα). − α − The inverse α-stable subordinator (Yα(t))t≥0 (see e.g. Bingham (1971)) is defined by Y (t) := inf u 0 : L (u) >t . α { ≥ α } We assume (Y (t)) to be independent of (N(t)) . For α (0, 1), the α t≥0 t≥0 ∈ fractional non-homogeneous Poisson process (FNPP) (Nα(t))t≥0 is defined as

Nα(t) := N(Yα(t)) = N1(Λ(Yα(t))) (2.1) (see Leonenko et al. (2017)). Note that the fractional homogeneous Poisson process (FHPP) is a special case of the non-homogeneous Poisson process with Λ(t) = λt, where λ(t) λ > 0 a constant. Recall that the density h (t, ) of ≡ α · Yα(t) can be expressed as (see e.g. Meerschaert and Straka, 2013; Leonenko and Merzbach, 2015) t t hα(t, x)= 1 gα 1 , x 0,t 0, (2.2) 1+ α α ≥ ≥ αx x where gα(z) is the density of Lα(1) given by ∞ 1 Γ(αk + 1) 1 g (z)= ( 1)k+1 sin(πkα) α π − k! zαk+1 Xk=1 The Laplace transform of hα can be given in terms of the Mittag-Leﬄer function ∞ h˜ (t,y)= e−xyh (t, x) dx = E ( ytα), (2.3) α α α − Z0 and for the FNPP the one-dimensional marginal distributions are given by ∞ Λ(u)x P(N (t)= x)= e−Λ(u) h (t,u) du. α x! α Z0 Alternatively, we can construct an non-homogeneous Poisson process as follows (see Gergely and Yezhow (1973)). Let ξ1, ξ2,... be a sequence of independent non-negative random variables with identical continuous distribution function F (t)= P(ξ t)=1 exp( Λ(t)),t 0. 1 ≤ − − ≥

3 Deﬁne ζ′ := max ξ ,...,ξ , n =1, 2,... n { 1 n} and ′ ′ κ = inf k N : ζ > ζκ , n =2, 3,... n { ∈ k n−1 } κ ′ with 1 = 1. Then, let ζn := ζκn . The resulting sequence ζ1, ζ2,... is strictly ′ ′ increasing, since it is obtained from the non-decreasing sequence ζ1, ζ2,... by omitting all repeating elements. Now, we deﬁne

∞ N(t) := sup k N : ζ t = n1 , t 0 { ∈ k ≤ } {ζn≤t<ζn+1} ≥ n=0 X where ζ0 = 0. By Theorem 1 in Gergely and Yezhow (1973), we have that (N(t))t≥0 is a non-homogeneous Poisson process with independent increments and Λ(t)k P(N(t)= k) = exp( Λ(t)) , k =0, 1, 2,.... − k! It follows via the time-change approach that the FNPP can be written as

∞ ∞ 1 a.s. 1 Nα(t)= n {ζn≤Yα(t)<ζn+1} = n {Lα(ζn)≤t

3. The FNPP as Cox process

Cox processes go back to Cox (1955) who proposed to replace the deterministic intensity of a Poisson process by a random one. In this section, we discuss the connection between FNPP and Cox processes.

Definition 1. Let (Ω, , P) be a probability space and (N(t))t≥0 be a point process adapted to a filtrationF ( N ) . (N(t)) is a Cox process if there Ft t≥0 t≥0 exist a right-continuous, increasing process (A(t))t≥0 such that, conditional on the filtration ( ) , where Ft t≥0 := N , = σ(A(t),t 0), (3.1) Ft F0 ∨Ft F0 ≥ then (N(t))t≥0 is a Poisson process with intensity dA(t). In particular we have by definition E[N(t) ]= A(t) and |Ft A(t)k P(N(t)= k )=e−A(t) , k =0, 1, 2,... |Ft k!

4 Remarks 1. 1. A Cox process N is said to be directed by A, if their relation is as in the above definition. Cox processes are also called ( ) -Cox pro- Ft t≥0 cess, doubly stochastic processes, conditional Poisson processesor( t)t≥0- conditional Poisson process. F 2. As both N and A are increasing processes, they are also of finite variation. The path t N(t,ω) induces positive measures dN and integration w.r.t. this measure7→ can be understood in the sense of a Lebegue-Stieltjes integral (see p. 28 in Jacod and Shiryaev (2003)). The same holds for the paths of A. 3. Definitions vary across the literature. The above definition can be compared to essentially equivalent definitions: in Brémaud (1981), 6.12 on p. 126 in Jacod and Shiryaev (2003), Definition 6.2.I on p. 169 in Daley and Vere-Jones (2008), where = R+ and Definition 6.6.2 on p.193 in Bielecki and Rutkowski (2002). X 4. Cox processes find applications in credit risk modelling. In this context A(t) is referred to as hazard process (see Bielecki and Rutkowski (2002)). In case of the FHPP, there exist characterizing theorems, for example, found in Yannaros (1994) and Grandell (1976) (Theorem 1 of Section 2.2). They use the fact that the FHPP is also a renewal process and allows for a characterization via the Laplace transform of the waiting time distributions. This has been worked out in detail in Section 2 in Leonenko et al. (2017). However, the theorem does not give any insight about the underlying filtration setting. This will become more evident from the following discussion concerning the general case of the FNPP.

In the non-homogeneous case, we cannot apply the theorems which charac- terize Cox renewal processes as the FNPP cannot be represented as a classical- renewal process. Therefore, we need to resort to Definition 1 for verification. It can be shown that the FNPP is a Cox process under a suitably constructed filtration. We will follow the construction of doubly stochastic processes given in Section 6.6 in Bielecki and Rutkowski (2002). Let ( Nα ) be the natural Ft t≥0 filtration of the FNPP (Nα(t))t≥0 Nα := σ( N (s): s t ). Ft { α ≤ } We assume the paths of the inverse α-stable subordinator to be known, i.e. := σ( Y (t),t 0 ). (3.2) F0 { α ≥ } We refer to this choice of initial σ-algebra as non-trivial initial history as opposed to the case of trivial initial history, which is 0 = ∅, Ω . The overall filtration ( ) is then given byF { } Ft t≥0 := Nα , (3.3) Ft F0 ∨Ft which is sometimes referred to as intrinsic history. If we choose a trivial initial history, the intrinsic history will coincide with the natural filtration of the FNPP. Proposition 1. Let the FNPP be adapted to the filtration ( ) as in (3.3) Ft with non-trivial initial history 0 := σ( Yα(t),t 0 ). Then the FNPP is a ( )-Cox process directed by (Λ(FY (t))) { . ≥ } Ft α t≥0

5 Proof. This follows from Proposition 6.6.7. on p. 195 in Bielecki and Rutkowski (2002). We give a similar proof: As (Y (t)) is -measurable we have α t≥0 F0 E[exp iu(N (t) N (s)) ] { α − α }|Fs = E exp iu(N (t) N (s)) Nα { α − α }|F0 ∨Fs N1 = E exp iu(N1(Λ(Yα(t))) N1(Λ(Yα(s)))) 0 (3.4) { − }|F ∨FΛ(Yα(s)) = E h[exp iu(N (Λ(Y (t))) N (Λ(Y (s)))) ]i (3.5) { 1 α − 1 α }|F0 = exp[Λ(Y (s), Y (t))(eiu 1)], α α − where in (3.4) we used the time-change theorem (see for example Thm. 7.4.I. p. 258 in Daley and Vere-Jones (2003)) and in (3.5) the fact that the standard Poisson process has independent increments. This means, conditional on ( ) , (N (t)) has independent increments and Ft t≥0 α (N (t) N (s)) Poi(Λ(Y (s), Y (t))) =d Poi(Λ(Y (t)) Λ(Y (s))). α − α |Fs ∼ α α α − α

Thus, (N(Yα(t))) is a Cox process directed by Λ(Yα(t)) by deﬁnition.

4. The FNPP and its compensator

The idenfication of the FNPP as a Cox process in the previous section allows us to determine the compensator of the FNPP. In fact, the compensator of a Cox process coincides with its directing process. From Lemma 6.6.3. p.194 in Bielecki and Rutkowski (2002) we have the result Proposition 2. Let the FNPP be adapted to the filtration ( ) as in (3.3) with Ft non-trivial initial history 0 := σ( Yα(t),t 0 ). Assume E[Λ(Yα(t))] < . Then the FNPP has -compensatorF { (A(t))≥ }, where A(t) := Λ(Y (t)), i.e.∞ Ft t≥0 α the stochastic process (M(t))t≥0 defined by M(t) := N(Yα(t)) Λ(Yα(t)) is a -martingale. − Ft 4.1. A central limit theorem Using the compensator of the FNPP, we can apply martingale methods in order to derive limit theorems for the FNPP. For the sake of completeness, we restate the definition of 0-stable convergence along with the theorem which will be used later. F

Deﬁnition 2. If (Xn)n∈N and X are R-valued random variables on a probability space (Ω, , P) and is a sub-σ-algebra of , then Xn X ( -stably) in distributionE if for allFB and all A (R)E with P(X ∂A→ ) = 0,F ∈F ∈B ∈

P( Xn A B) P( X A B) { ∈ } ∩ −−−−→n→∞ { ∈ } ∩ (see Deﬁnition A.3.2.III. in Daley and Vere-Jones (2003)). Note that -stable converges implies weak convergence/convergence in distribution. WeF can derive a central limit theorem for the FNPP using Corollary 14.5.III. in Daley and Vere-Jones (2003) which we state here as a lemma for convenience.

6 Lemma 1. Let N be a simple point process on R+, ( t)t≥0-adapted and with continuous ( ) -compensator A. Set F Ft t≥0 T X := f (u)[dN(u) dA(u)]. T T − Z0 Suppose for each T > 0 an ( ) -predictable process f (t) is given such that Ft t≥0 T T 2 2 BT = [fT (u)] dA(u) > 0. Z0

Then the randomly normed integrals XT /BT converge 0-stably to a standard normal variable W N(0, 1). F ∼ Note that the above integrals are well-deﬁned as explained in Point 2 in Remarks 1. The above theorem allows us to show the following result for the FNPP.

Proposition 3. Let (N(Yα(t)))t≥0 be the FNPP adapted to the ﬁltration ( ) as deﬁned in Section 3. Then, Ft t≥0

N(Yα(T )) Λ(Yα(T )) − W N(0, 1) 0-stably. (4.1) Λ(Yα(T )) −−−−→T →∞ ∼ F p Proof. First note that the compensator A(t) := Λ(Yα(t)) is continuous in t. Let f (u) C a constant, then T ≡ T T B2 = [f (u)]2dA(u)= C2 (Λ(Y (T )) Λ(Y (0))) T T T α − α Z0 = C2 Λ(Y (T )) > 0, T > 0 T α ∀ and

T X := f (u)[dN(Y (u)) dA(u)] T T α − Z0 = C [N(Y (T )) A(T ) N(Y (0)) + A(0)] T α − − α = C [N(Y (T )) A(T )] = C [N(Y (T )) Λ(Y (T ))]. T α − T α − α It follows from Theorem 1 above that X C [N(Y (T )) Λ(Y (T ))] T = T α − α B 2 T CT Λ(Yα(T )) N(Yα(T )) Λ(Yα(T )) = p − W N(0, 1) 0-stably. Λ(Yα(T )) −−−−→T →∞ ∼ F p

4.2. Limit α 1 → In the following, we give a more rigorous proof for the limit α 1 in Section 3.2(ii) in Leonenko et al. (2017). →

7 Proposition 4. Let the FNPP be adapted to the ﬁltration ( ) as in (3.3) with Ft non-trivial initial history 0 := σ( Yα(t),t 0 ). Let (Nα(t))t≥0 be the FNPP as deﬁned in (2.1). Then,F we have{ the limit≥ }

J1 Nα N in D([0, )). −−−→α→1 ∞

Proof. By Proposition 2 we see that (Λ(Yα(t)))t≥0 is the compensator of (Nα(t))t≥0. According to Theorem VIII.3.36 on p. 479 in Jacod and Shiryaev (2003) if suf- ﬁces to show P Λ(Yα(t)) Λ(t). −−−→α→1 We can check that the Laplace transform of the density of the inverse α-stable subordinator converges to the Laplace transform of the delta distribution:

α→1 h ( ,y) (s,y)= E ( ysα) e−ys = δ ( y) (s,y). (4.2) L{ α · } α − −−−→ L{ 0 ·− } We may take the limit as the power series representation of the (entire) Mittag- Leﬄer function is absolutely convergent. Thus (4.2) implies

d Yα(t) t t R+. −−−→α→1 ∀ ∈ As convergence in distribution to a constant automatically improves to convergence in probability, we have

P Yα(t) t t R+. −−−→α→1 ∀ ∈ By the continuous mapping theorem, it follows that

P Λ(Yα(t)) Λ(t) t R+, −−−→α→1 ∀ ∈ which concludes the proof.

5. Regular variation and scaling limits

In this section we will work with the trivial initial filtration setting ( = F0 ∅, Ω ), i.e. t is assumed to be the natural filtration of the FNPP. In this setting,{ } the FNPPF can generally not be seen as a Cox process and although the compensator of the FNPP does exist, it is difficult to give a closed form expression for it. Instead, we follow the approach of results given in Grandell (1976), Serfozo (1972a), Serfozo (1972b), which require conditions on the function Λ. Recall that a function Λ is regularly varying with index β R if ∈ Λ(xt) xβ, x> 0. (5.1) Λ(t) −−−→t→∞ ∀

Under the mild condition of measurability, one can show that the above limit is quite general in the sense that if the quotient of the right hand side of (5.1) converges to a function x g(x), g has to be of the form xβ (see Thm. 1.4.1 in Bingham et al. (1989)).7→

8 Example 1. We check whether typical rate functions (taken from Remark 2 in Leonenko et al. (2017)) fulﬁll the regular variation condition. (i) Weibull’s rate function t c c t c−1 Λ(t)= , λ(t)= , c 0,b> 0 b b b ≥ is regulary varying with index c. This can be seen as follows Λ(xt) (xt)c = = xc, x> 0. Λ(t) tc ∀

(ii) Makeham’s rate function c c Λ(t)= ebt + µt, λ(t)= cebt + µ, c> 0,b> 0, µ 0 b − b ≥ is not regulary varying, since Λ(xt) (c/b)ebxt (c/b)+ µxt (c/b)ebt(x−1) (c/b)e−bt + µxte−bt = − = − Λ(t) (c/b)ebt (c/b)+ µt (c/b) (c/b)e−bt + µte−bt − − 0 if x< 1 t→∞ 1 if x =1  −−−→ + if x> 1  ∞ does not fulﬁll (5.1).  △ In the following, the condition that Λ is regularly varying is useful for proving limit results. We will ﬁrst show a one-dimensional limit theorem before moving on to the functional analogue.

5.1. A one-dimensional limit theorem

Theorem 5. Let the FNPP (Nα(t))t≥0 be deﬁned as in Equation (2.1). Sup- pose the function t Λ(t) is regularly varying with index β R. Then the following limit holds7→ for the FNPP: ∈

Nα(t) d β (Yα(1)) . (5.2) Λ(tα) −−−→t→∞ Proof. We will ﬁrst show that the characteristic function of the random variable on the left hand side of (5.2) converges to the characteristic function of the right hand side. By self-similarity of Yα we have

d α N1(Λ(Yα(t))) = N1(Λ(t Yα(1))).

Nα(t) Therefore, it follows for the characteristic function of Z(t) := Λ(tα) that α −1 ϕ(t) := E[exp(iuZ(t))] = E[exp(iuΛ(t ) N1(Λ(Yα(t))))] α −1 α = E[exp(iuΛ(t ) N1(Λ(t Yα(1))))] ∞ α −1 α = E[exp(iuΛ(t ) N1(Λ(t x)))]hα(1, x) dx (5.3) 0 Z ∞ α −1 = exp(Λ(tαx)(eiuΛ(t ) 1))h (1, x) dx, (5.4) − α Z0

9 where we used a conditioning argument in (5.3), x h (1, x) is the density 7→ α function of the distribution of Yα(1). In the last step in (5.4) we may insert the characteristic function of a Poisson distributed random variable with parameter Λ(tαx) evaluated at the point uΛ(tα)−1. In order to pass to the limit, we need to justify that we may exchange integration and limit. It can be observed that the integrand is dominated by an integrable function independent of t. By Jensen’s inequality E[exp(iuΛ(tα)−1 N (Λ(tαx)))]h (1, x) 1 α | E α −1 α [ exp(iuΛ(t ) N1(Λ(t x))) ]hα(1, x) hα(1, x) ≤ | | ≤ This allows us to use the dominated convergence theorem to get

∞ − α iuΛ(tα ) 1 lim ϕ(t) = lim exp(Λ(t x)(e 1))hα(1, x) dx t→∞ t→∞ − Z0 ∞ − α iuΛ(tα ) 1 = lim exp(Λ(t x)(e 1)) hα(1, x) dx. (5.5) 0 t→∞ − Z h i We are left with calculating the limit in the square bracket in (5.5). To this α −1 end, consider a power series expansion of eiuΛ(t ) to observe that

∞ − k α iuΛ(tα ) 1 α (iu) exp Λ(t x)(e 1) = exp Λ(t x) α k − Λ(t ) k!!! Xk=1 iu Λ(tαx) 1 = exp + Λ(tαx) , 1! Λ(tα) O Λ(tα)2 ! xβ 0 −−−→t→∞ −−−→t→∞ where we have used that Λ is regularly varying| {z with} | index β {zin the last} step. Inserting this result into (5.5) yields ∞ β lim ϕ(t)= exp iux hα(1, x)dx t→∞ 0 Z β = E[eiu(Yα(1)) ]. Applying L´evy’s continuity theorem concludes the proof. Remark 2. The above result can be shown alternatively using Theorem 3.4 in Serfozo (1972a) or Theorem 1 on pp. 69-70 in Grandell (1976). The limit α β distribution of Nα(t)/Λ(t ) is the sum of the limit distribution (Yα(1)) of the inner process Λ(Yα(t)) and a normal distribution (the limit of the outer process, the Poisson process). The variance of the normal distribution is determined by the norming constants in the inner process limit. In our case the variance is 0 β and we are left with (Yα(1)) as limit of the overall process. Remark 3. As a special case of the theorem we get for Λ(t)= λt, for constant λ> 0 Λ(xt) = x1 Λ(t) which means Λ is regularly varying with index β = 1. It follows that

N1(λYα(t)) d Yα(1). λtα −−−→t→∞

10 This is in accordance to the scaling limit given in Cahoy et al. (2010) who showed N1(λYα(t)) N1(λYα(t)) d = α Γ(1 + α)Yα(1). E λt t→∞ [N1(λYα(t))] Γ(1+α) −−−→

5.2. A functional limit theorem The one-dimensional result in Theorem 5 can be extended to a functional limit theorem. In the following we consider the Skorohod space ([0, )) en- D ∞ dowed with a suitable topology (we will focus on the J1 and M1 topology). For more details see Meerschaert and Sikorskii (2012).

Theorem 6. Let the FNPP (Nα(t))t≥0 be deﬁned as in Equation (2.1). Sup- pose the function t Λ(t) is regularly varying with index β R. Then the following limit holds7→ for the FNPP: ∈

Nα(tτ) J1 β [Yα(τ)] . (5.6) Λ(tα) −−−→t→∞ τ≥0 τ≥0 Remark 4. As the limit process has continuous paths the mode of convergence improves to local uniform convergence. Also in this theorem, we will denote the homogeneous Poisson process with intensity parameter λ = 1 with N1. In order to proof the Theorem we need Theorem 2 on p. 81 in Grandell (1976), which we will state here for convenience. Theorem 7. Let Λ¯ be a stochastic process in ([0, )) with Λ(0)¯ = 0 and let N = N (Λ)¯ be the corresponding doubly stochasticD process.∞ Let a ([0, )) 1 ∈ D ∞ with a(0) = 0 and t bt a positive regularly varying function with index ρ> 0 such that 7→ a(t) κ [0, ) and bt −−−→t→∞ ∈ ∞ ¯ Λ(tτ) a(tτ) J1 − (S(τ))τ≥0, b −−−→t→∞ t τ≥0 where S is a stochastic process in ([0, )). Then D ∞

N(tτ) a(tτ) J1 − (S(τ)+ h(B(τ)))τ≥0, b −−−→t→∞ t τ≥0 2ρ where h(τ) = κτ and (S(t))t≥0 and (B(t))t≥0 are independent. (B(t))t≥0 is the standard Brownian motion in ([0, )). D ∞ Proof of Thm. 6. We apply Theorem 7 and choose a 0 and b = Λ(tα). Then ≡ t it follows that κ = 0 and it can be checked that bt is regularly varying with index αβ: α α bxt Λ(x t ) αβ = α x bt Λ(t ) −−−→t→∞ by the regular variation property in (5.1). We are left to show that

Λ(Yα(tτ)) J1 β Λ˜ t(τ) := [Yα(τ)] . (5.7) Λ(tα) −−−→t→∞ τ≥0 τ≥0 11 This can be done by following the usual technique of first proving convergence of the finite-dimensional marginals and then tightness of the sequence in the Skorohod space ([0, )). Concerning theD convergence∞ of the finite-dimensional marginals we show convergence of their respective characteristic functions. Let t> 0 be fixed at first, Rn Rn τ = (τ1, τ2,...,τn) + and , denote the scalar product in . Then, we can write the characteristic∈ functionh· ·i of the joint distribution of the vector

Λ(tαY (τ)) Λ(tαY (τ )) Λ(tαY (τ )) Λ(tαY (τ )) α = α 1 , α 2 ,..., α n Rn Λ(tα) Λ(tα) Λ(tα) Λ(tα) ∈ + as Λ(Y (tτ)) Λ(tαY (τ)) ϕ (u) := E exp i u, α = E exp i u, α t Λ(tα) Λ(tα) Λ(tαx) = exp i u, α hα(τ, x)dx Rn Λ(t ) Z + n α Λ(t xk) = exp iuk α hα(τ1,...,τn; x1,...,xn)dx1 ... dxn Rn Λ(t ) + " # Z kY=1 where u Rn and h (τ, x) = h (τ , τ ,...,τ ; x , x ...,x ) is the density of ∈ α α 1 2 n 1 2 n the joint distribution of (Yα(τ1), Yα(τ2),...,Yα(τn)). We can ﬁnd a dominating function by the following estimate:

Λ(tαx) exp i u, h (τ, x) h (τ, x). Λ(tα) α ≤ α

The upper bound is an integrable function which is independent of t. By dominated convergence we may interchange limit and integration:

Λ(tαx) lim ϕn(u) = lim exp i u, α hα(τ, x)dx t→∞ t→∞ Rn Λ(t ) Z + Λ(tαx) = lim exp i u, α hα(τ, x)dx Rn t→∞ Λ(t ) Z + β β = exp i u, x hα(τ, x)dx = E[exp(i u, (Yα(τ)) )], Rn h i Z + where in the last step we used the continuity of the exponential function and the scalar product to calculate the limit. By Lévy’s continuity theorem we may conclude that for n N ∈ Λ(Yα(tτk)) d β [Yα(τk)] . Λ(tα) −−−→t→∞ k=1,...,n k=1,...,n In order to show tightness, first observe that for fixed t both the stochastic β process Λ˜ t on the left hand side and the limit candidate ([Yα(τ)] )τ≥0 have increasing paths. Moreover, the limit candidate has continuous paths. Therefore we are able to invoke Thm. VI.3.37(a) in Jacod and Shiryaev (2003) to ensure tightness of the sequence (Λ˜ t)t≥0 and thus the assertion follows.

12 By applying the transformation theorem for probability densities to (2.2), β we can write for the density hα(t, ) of the one-dimensional marginal of the limit β · process ([Yα(t)] )t≥0 as 1 hβ (t, x)= x1/β−1h (t, x1/β ) α β α 1 t t = x1/β−1 g β αx1/β(1+1/α) α y1/(αβ) t t = g . (5.8) αβx1+1/(αβ) α y1/(αβ)

Note that this is not the density of Yαβ(t). A further limit result can be obtained for the FHPP via a continuous mapping argument.

Proposition 8. Let (N1(t))t≥0 be a homogeneous Poisson process and (Yα(t))t≥0 be the inverse α-stable subordinator. Then

N1(Yα(t)) λYα(t) J1 − (B(Yα(t)))t≥0, √ −−−−→λ→∞ λ t≥0 where (B(t))t≥0 is a standard Brownian motion. Proof. The classical result

N1(t) λt J1 − (B(t))t≥0 √ −−−−→λ→∞ λ t≥0 can be shown by using that (N(t) λt) is a martingale. As (B(t)) has − t≥0 t≥0 continuous paths and (Yα(t))t≥0 has increasing paths we may use Theorem 13.2.2 in Whitt (2002) to obtain the result. The above proposition can be compared with Lemma 2 in the next section and a similar continuous mapping argument is applied in the proof of Theorem 10.

6. The fractional compound Poisson process

Let X1,X2,... be a sequence of i.i.d. random variables. The fractional compound Poisson process is deﬁned analogouly to the standard Poisson process where the Poisson process is replaced by a fractional one:

Nα(t)

Zα(t) := Xk, Xk=1 0 where k=1 Xk := 0. The process Nα is not necessarily independent of the Xi’s unless stated otherwise. StableP laws can be deﬁned via the form of their characteristic function.

13 Deﬁnition 3. A random variable S is said to have stable distribution if there are parameters 0 < α 2, σ 0, 1 β 1 and µ R such that its characteristic function has≤ the following≥ − form:≤ ≤ ∈ exp σα θ α 1 iβ sign(θ) tan πα + iµθ if α =1, E[exp(iθS)] = − | | − 2 6 exp σ θ 1 + iβ 2 sign(θ) ln( θ ) + iµθ if α =1 − | | π | | (see Deﬁnition 1.1.6 in Samorodnitsky and Taqqu (1994)). We will assume a limit result for the sequence of partial sums without time-change

n Sn := Xk, kX=1 usually a stable limit, i.e. there exist sequences (an)n∈N and (bn)n∈N and and a random variable following a stable distribution S such that

d S¯n := anSn bn S. − −−−−→n→∞ (for details see for example Chapter XVII in Feller (1971)). In other words the distribution of the Xk’s is in the domain of attraction of a stable law. In the following we will derive limit theorems for the fractional compound Poisson process. In Section 6.2, we assume Nα to be independent of the Xk’s and use a continuous mapping theorem argument to show functional convergence w.r.t. a suitable Skorohod topology. A corresponding one-dimensional limit theorem would follow directly from the functional one. However, in the special case of Nα being a FHPP, using Anscombe type theorems in Section 6.1 allows us to drop the independence assumption between Nα and the Xk’s and thus strengthen the result for the one-dimensional limit.

6.1. A one-dimensional limit result The following theorem is due to Anscombe (1952) and can be found slightly reformulated in Richter (1965). Theorem 9. We assume that the following conditions are fulﬁlled:

(i) The sequence of random variables Rn such that

d Rn R, −−−−→n→∞ for some random variable R. (ii) Let the family of integer-valued random variables N(t) be relatively stable, i.e. for a real-valued function ψ with ψ(t) < it holds that −−−→t→∞ ∞ N(t) P 1. ψ(t) −−−→t→∞

(iii) (Uniform continuity in probability) For every ε> 0 and η > 0 there exists a c = c(ε, η) and a t = t (ε, η) such that for all t t 0 0 ≥ 0

P max Rm Rt >ε < η. m:|m−t|

14 Then, d RN(t) R. −−−→t→∞

We would like to use the above theorem for Rn = Sn. Indeed, condition (i) follows from the assumption that the law of X1 lies in the domain of attraction of a stable law. It is readily veriﬁed in Theorem 3 in Anscombe (1952) that (Sn) fulﬁlls the condition (iii). Concerning the condition (ii), note that the required convergence in probability is stronger than the convergence in distribution we have derived in the previous sections for the FNPP. Nevertheless, in the special case of the FHPP, we can improve the mode of convergence.

λ Lemma 2. Let Nα be a FHPP, i.e. Λ(t)= λt in (2.1). Then with C := Γ(1+α) it holds that N (t) α P 1. Ctα −−−→t→∞ Proof. According to Proposition 4.1 from Di Crescenzo et al. (2016) we have the result that for ﬁxed t> 0 the convergence

N1(λYα(t)) N1(λYα(t)) L1 = α 1 (6.1) E λt λ→∞ [N1(λYα(t))] Γ(1+α) −−−−→ holds and therefore also in probability. It can be shown by using the fact that the moments and the waiting time distribution of the FHPP can be expressed in terms of the Mittag-Leﬄer function. Let ε> 0. We have

α N1(λYα(t)) N1(λt Y (1)) lim P 1 >ε = lim P α 1 >ε (6.2) t→∞ Ctα − t→∞ λt − Γ(1+α) !

P N1(τY (1)) = lim τ·1α 1 >ε =0, (6.3) τ→∞ − ! Γ(1+α)

where in (6.2) we used the self-similarity property of Yα and in (6.3) we applied (6.1) with t = 1. By applying Lemma 2 condition (ii) is satisﬁed and it follows that

Nα(t) ¯ d SN (t) = aN (t) Xk bN (t) S α α − α −−−→t→∞ Xk=1 and (see Theorem 3.6 in Gut (2013))

Nα(t) ¯ d SN (t) = a⌊Ct⌋ Xk b⌊Ct⌋ = a⌊Ct⌋Zα(t) b⌊Ct⌋ S. α − − −−−→t→∞ Xk=1

Note that this convergence result does not require Nα to be independent of the Xk’s. The above derivation also works for mixing sequences X1,X2,... instead of i.i.d. (see Cs¨org˝oand Fischler (1973) for a generalisation of Anscombe’s theorem for mixing sequences).

15 6.2. A functional limit theorem

Theorem 10. Let the FNPP (Nα(t))t≥0 be deﬁned as in Equation (2.1) and suppose the function t Λ(t) is regularly varying with index β R. Moreover 7→ ∈ let X1,X2,... be i.i.d. random variables independent of Nα. Assume that the law of X1 is in the domain of attraction of a stable law, i.e. there exist sequences (an)n∈N and (bn)n∈N and a stable L´evy process (S(t))t≥0 such that for

⌊nt⌋ S¯ (t) := a X b n n k − n Xk=1 it holds that J1 (S¯n(t))t≥0 (S(t))t≥0. (6.4) −−−−→n→∞

Then the fractional compound Poisson process Z(t) := SNα(t) fulﬁlls following limit: J1 β (cnZ(nt))t≥0 S [Yα(t)] , −−−−→n→∞ t≥0

where cn = a⌊Λ(n)⌋. Proof. The proof follows the technique proposed by Meerschaert and Scheﬄer (2004): By Theorem 6 we have

Nα(tτ) J1 β [Yα(τ)] . Λ(tα) −−−→t→∞ τ≥0 τ≥0 By the independence assumptions we can combine this with (6.4) to get

α α −1 J1 β a⌊Λ(nα)⌋S(Λ(n )) bn, [Λ(n )] Nα(nt) (S(t), [Yα(t)] )t≥0 − t≥0 −−−−→n→∞ in the space ([0, ), R [0, )). Note that ([Y (t)]β ) is non-decreasing. D ∞ × ∞ α t≥0 Moreover, due to independence the L´evy processes (S(t))t≥0 and (Dα(t))t≥0 do not have simultaneous jumps (for details see Becker-Kern et al. (2004) and more generally Cont and Tankov (2004)). This allows us to apply Theorem 13.2.4 in Whitt (2002) to get the assertion by a continuous mapping argument since the composition mapping is continuous in this setting.

7. Numerical experiments

7.1. Simulation methods In the special case of the FHPP, the process is simulated by sampling the waiting times Ji of the overall process N(Yα(t)), which are Mittag-Leﬄer distributed (see Equation (1.1)). Direct sampling of the waiting times of the FHPP can be done via a transformation formula due to Kozubowski and Rachev (1999)

1 sin(απ) 1/α J = log(U) cos(απ) , 1 −λ tan(απV ) − where U and V are two independent random variables uniformly distributed on [0, 1]. For futher discussion and details on the implementation see Fulger et al.

16 (2008) and Germano et al. (2009).

As the above method is not applicable for the FNPP, we draw samples of Yα(t) ﬁrst before sampling N. The Laplace transform w.r.t. the time variable of Yα(t) is given by

∞ e−sth (t, x) dt = sα−1 exp( xsα). α − Z0

We evaluate the density hα by inverting the Laplace transform numerically using the Post-Widder formula (Post (1930) and Widder (1941)): Theorem 11. If the integral

∞ f¯(s)= e−suf(u)du Z0 converges for every s>γ, then

( 1)n n n+1 n f(t) = lim − f¯(n) , n→∞ n! t t for every point t> 0 of continuity of f(t) (cf. p. 37 in Cohen (2007)).

This evaluation of the density function allows us to sample Yα(t) using dis- crete inversion.

7.2. Numerical results Figure 1 shows the shape and time-evolution of the densities for diﬀerent values of α. As Yα is an increasing process, the densities spread to the right hand side as time passes. We conducted a small Monte Carlo simulation in order to illustrate the one- dimensional convergence results of Theorem 1 and Theorem 5. In Figures 2, 3 and 4, we can see that the simulated values for the probability density x ϕα(t, x) of [N(Yα(t)) Λ(Yα(t))]/ Λ(Yα(t)) approximate the density of a standard7→ normal distribution− for increasing time t. In a similar manner, Figure p α 5 depicts how the probability density function x φα(t, x) of Nα(t)/Λ(t ) ap- β 7→ proximates the density of (Yα(t)) given in (5.8), where Λ has regular variation index β =0.7.

8. Summary and outlook

Due to the non-homogeneous component of the FNPP, it is not surprising that analytical tractability needed to be compromised in order to derive anal- ogous limit theorems. Most noteably, the lack of a renewal representation of the FNPP compared to its homogeneous version lead us to require additional conditions on the underlying ﬁltration structure or rate function Λ. The result in Proposition 4 partly answered an open question that followed after Theorem 1 in Leonenko et al. (2017) concerning the limit α 1. Futher research will be directed towards the implications of the→ limit results for estmation techniques.

17 1.2 0.6 0.5 α = 0.1 1 0.5 α = 0.6 0.4 α = 0.9 0.8 0.4 ) ) ) 0.3 , x , x , x 0.6 0.3 (1 (10 (40 α α α h

h h 0.2 0.4 0.2

0.1 0.2 0.1

0 0 0 0 5 10 0 10 20 30 0 50 100 x x x

Figure 1: Plots of the probability densities x 7→ hα(t, x) of the distribution of the inverse α-stable subordinator Yα(t) for diﬀerent parameter α = 0.1, 0.6, 0.9 indicating the time- evolution: the plot on the left is generated for t = 1, the plot in the middle for t = 10 and the plot on the right for t = 40.

0.6 0.45 0.4

0.4 0.35 0.5 0.35 0.3 0.4 0.3 )

) ) 0.25 , x , x , x

3 9 0.25 12 0.3 0.2 (10 (10 (10 1 1 1 . .

0.2 . 0 0 0 ϕ ϕ ϕ 0.15 0.2 0.15 0.1 0.1 0.1 0.05 0.05

0 0 0 -10 0 10 20 -5 0 5 10 -5 0 5 10 x x x

Figure 2: The red line shows the probability density function of the standard normal distribution, the limit distribution according to Theorem 1. The blue histograms depict samples of size 104 of the right hand side of (4.1) for diﬀerent times t = 10, 109, 1012 to illustrate convergence to the standard normal distribution.

18 0.6 0.45 0.45

0.4 0.4 0.5 0.35 0.35

0.4 0.3 0.3 ) ) ) , x

, x 0.25 0.25 , x

(1 0.3 (10 6 (100 . 6 . 6 0 .

0 0.2 0.2 0 ϕ ϕ ϕ 0.2 0.15 0.15

0.1 0.1 0.1 0.05 0.05

0 0 0 -5 0 5 10 -5 0 5 10 -5 0 5 x x x

Figure 3: The red line shows the probability density function of the standard normal distribution, the limit distribution according to Theorem 1. The blue histograms depict samples of size 104 of the right hand side of (4.1) for diﬀerent times t = 1, 10, 100 to illustrate convergence to the standard normal distribution.

0.6 0.4 0.45

0.35 0.4 0.5 0.35 0.3 0.4 0.3 0.25 ) ) )

, x , x 0.25 , x

(1 0.3 0.2 (10 (20 9 . 9 9 . . 0

0 0 0.2 ϕ ϕ 0.15 ϕ 0.2 0.15 0.1 0.1 0.1 0.05 0.05

0 0 0 -5 0 5 10 -5 0 5 -5 0 5 x x x

Figure 4: The red line shows the probability density function of the standard normal distribution, the limit distribution according to Theorem 1. The blue histograms depict samples of size 104 of the right hand side of (4.1) for diﬀerent times t = 1, 10, 20 to illustrate convergence to the standard normal distribution.

19 1.2 1.5 1.2

1 1 ) 0.8 1 ) 0.8 ) , x , x , x 3 0.6 0.6 (10 (10 (100 9 . 9 9 . . 0 0 0 φ φ 0.4 φ 0.5 0.4

0.2 0.2

0 0 0 0 2 4 0 2 4 0 2 4 x x x

Figure 5: Red line: probability density function φ of the distribution of the random variable 0.7 4 (Y0.9(1)) , the limit distribution according to Theorem 5. The blue histogram is based on 10 samples of the random variables on the right hand side of (5.2) for time points t = 10, 100, 103 to illustrate the convergence result.

Acknowledgements

N. Leonenko was supported in particular by Cardiﬀ Incoming Visiting Fel- lowship Scheme and International Collaboration Seedcorn Fund and Australian Research Council’s Discovery Projects funding scheme (project number DP160101366). E. Scalas and M. Trinh were supported by the Strategic Development Fund of the University of Sussex.

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