Some Functional (Hölderian) Limit Theorems and Their Applications

Some functional (H¨olderian)limit theorems and their applications (I)

Alfredas Raˇckauskas

Vilnius University

Outils Statistiques et Probabilistes pour la Finance Universit´ede Rouen June 1–5, Rouen

(Rouen 2015) Some functional limit theorems 1 / 94 2 To demonstrate possible applications of the weak invariance principle. This include detection of changed segment in a sample, e.g. epidemic change of a mean, of a variance and some other parameters.

Aims of the lectures

(Rouen 2015) Some functional limit theorems 2 / 94 Aims of the lectures

1 To introduce the weak invariance principle in a H¨olderian framework (Lamperti’s type IP) by considering large family of random structures including independent identically distributed random variables, linear processes, random ﬁelds and others. 2 To demonstrate possible applications of the weak invariance principle. This include detection of changed segment in a sample, e.g. epidemic change of a mean, of a variance and some other parameters.

(Rouen 2015) Some functional limit theorems 2 / 94 Aims of the lectures

(Rouen 2015) Some functional limit theorems 2 / 94 Outline of the lecture Lamperti’s invariance principle

1 Introduction

2 H¨olderspaces

3 Weak convergence and tightness

4 Lamperti’s invariance principle

5 Adaptive FCLT

6 FCLT for triangular arrays

(Rouen 2015) Some functional limit theorems 3 / 94 Motto of the lecture

Theory without practice is empty Practice without theory is blind (adapted from Immanuel Kant)

(Rouen 2015) Some functional limit theorems 4 / 94 Introduction

First example

Suppose we have a population (e.g., monthly earnings, pass-rates etc.) whose mean, µ, is unknown. In order to learn something about µ, one takes a large independent sample X1,..., Xn from the population under consideration, and constructs the sample average 1 X := (X + ··· + X ). n n 1 n

By the strong law of large numbers, X n ≈ µ. In order to ﬁnd a more quantitative estimate one can use the central limit theorem (CLT) which is most rudimentary example from which one could start the story named ,,invariance principle”.

(Rouen 2015) Some functional limit theorems 5 / 94 CLT [classical] If σ2 ∈ (0, ∞), then

1 D √ (Sn − nµ) −−−→ N (0, 1), nσ n→∞

where −−−→D means convergence in distribution and N (m, σ2) denotes the n→∞ normal random variable with mean m and variance σ2.

−−−→D n→∞

Let X1, X2,... be independent identically distributed (i.i.d.) random 2 2 variables, EX1 = µ, EX1 = σ . Denote their partial sums by

k X S0 = 0, Sk = Xj , j ≥ 1. j=1

(Rouen 2015) Some functional limit theorems 6 / 94 −−−→D n→∞

Let X1, X2,... be independent identically distributed (i.i.d.) random 2 2 variables, EX1 = µ, EX1 = σ . Denote their partial sums by

k X S0 = 0, Sk = Xj , j ≥ 1. j=1

CLT [classical] If σ2 ∈ (0, ∞), then

1 D √ (Sn − nµ) −−−→ N (0, 1), nσ n→∞ where −−−→D means convergence in distribution and N (m, σ2) denotes the n→∞ normal random variable with mean m and variance σ2.

(Rouen 2015) Some functional limit theorems 6 / 94 Let X1, X2,... be independent identically distributed (i.i.d.) random 2 2 variables, EX1 = µ, EX1 = σ . Denote their partial sums by

k X S0 = 0, Sk = Xj , j ≥ 1. j=1

CLT [classical] If σ2 ∈ (0, ∞), then

1 D √ (Sn − nµ) −−−→ N (0, 1), nσ n→∞ where −−−→D means convergence in distribution and N (m, σ2) denotes the n→∞ normal random variable with mean m and variance σ2.

−−−→D n→∞

(Rouen 2015) Some functional limit theorems 6 / 94 By this CLT we have for a < b ∈ R Z b √ 1 −x2/2 lim P( n(X n − µ)/σ ∈ (a, b)) = √ e dx. n→∞ 2π a One can then use this to derive approximate conﬁdence bounds for µ. Since P(N (0, 1) ∈ [−1.96, 1.96]) ≈ 0.95, we deduce

1.96 1.96 P(µ ∈ (X − √ , X + √ ) ≈ 0.95 n σ n n σ n provided the sample size n is ,,large enough”.

(Rouen 2015) Some functional limit theorems 7 / 94 This was precisely asymptotic analysis of tn that initiated the so-called self-normalized CLT.

If σ2 is unknown we can use its estimator

n 1 X σ2 = (X − X )2 b n k n k=1 and explore the so called Student’s t-statistics √ n(X n − µ) tn = . σb

(Rouen 2015) Some functional limit theorems 8 / 94 If σ2 is unknown we can use its estimator

n 1 X σ2 = (X − X )2 b n k n k=1 and explore the so called Student’s t-statistics √ n(X n − µ) tn = . σb

This was precisely asymptotic analysis of tn that initiated the so-called self-normalized CLT.

(Rouen 2015) Some functional limit theorems 8 / 94 Recall that X1 belongs to the domain of attraction of a normal distribution (denoted by X1 ∈ DAN) if there exists a norming sequence bn ↑ ∞ such that −1 D b Sn −−−→ N (0, 1). n n→∞

Theorem (Selfnormalized CLT [Gin´e,G¨otze,Mason (1997)])

Let X1,..., Xn be iid random variables. Then the convergence

Sn D q −−−→ N (0, 1) 2 2 n→∞ X1 + ··· + Xn holds if and only if X1 ∈ DNA and EX1 = 0.

(Rouen 2015) Some functional limit theorems 9 / 94 More precisely we have a sample X1, X2,..., Xn and we wish to know whether there is a k∗, 1 < k∗ < n, such that

EX1 = ··· = EXk∗ = µ,

0 EXk∗+1 = ··· = Xn = µ Figure: Annual temperature and µ 6= µ0.

Second example

Now suppose you are drawing samples as time passes, and wish to know if the mean of the underlying population (e.g., annual temperature) has changed over time.

(Rouen 2015) Some functional limit theorems 10 / 94 Second example

Now suppose you are drawing samples as time passes, and wish to know if the mean of the underlying population (e.g., annual temperature) has changed over time. More precisely we have a sample X1, X2,..., Xn and we wish to know whether there is a k∗, 1 < k∗ < n, such that

EX1 = ··· = EXk∗ = µ,

0 EXk∗+1 = ··· = Xn = µ Figure: Annual temperature and µ 6= µ0.

(Rouen 2015) Some functional limit theorems 10 / 94 The simplest way to construct a required test is to take two parts of the given sample, one X1,..., Xk and the second one Xk+1,..., Xn, and to compare their sample averages 1 1 (X + ··· + X ) and (X + ··· + X ). k 1 k n − k k+1 n Calculating their diﬀerence gives the quantity

k n h X i T (k) := X − kX n k(n − k) j n j=1 which should be large provided the equality of means before and after the moment k∗ = k fails. ∗ As the point of change k is not known one has to look at Tn(k) for all possible k = 1,..., n.

(Rouen 2015) Some functional limit theorems 11 / 94 Thus, it turns out that one needs to investigate the quantity

Tn := max (Sk − kX n). 1≤k≤n

This is CUSUM test statistics which is one of the most frequently used method to detect change points. Actually it might be easier to start with

Mn := max (Sj − jµ), n ≥ 1 1≤j≤n and ask the question whether there is a limit distribution for Mn as n → ∞ under the assumption that X1,..., Xn all have the same distribution? It turns out that the answer is positive, and involves an invariance principle.

(Rouen 2015) Some functional limit theorems 12 / 94 Erd¨osand Kac described their method of proof as follows: ,,The proof of all these theorems follow the same pattern. It is ﬁrst proved that the limiting distribution exists and is independent of the distribution of the Xi ’s; then the distribution of Xi ’s is chosen conveniently so that the limiting distribution can be calculated explicitly.” Perhaps the story ,,invariance principle” have realy started with their providence.

Erd¨os1 and Kac2contribution

The limit behavior of Mn = max1≤j≤n(Sj − jµ) was precisely the question that were interested Erd¨osand Kac (1946) in. They found

r Z x √ 2 −y 2/2 lim P( max (Sj − jµ) ≤ nx) = e dy n→∞ 1≤j≤n π 0

for all x ≥ 0, provided X1, X2,... are iid with mean µ and var(X1) = 1.

1Paul Erd¨os(26 March 1913 – 20 September 1996) was a Jewish-Hungarian mathematician 2Mark Kac (3 August 1914 – 26 October 1984) was a Polish American mathematician (Rouen 2015) Some functional limit theorems 13 / 94 Perhaps the story ,,invariance principle” have realy started with their providence.

Erd¨os1 and Kac2contribution

The limit behavior of Mn = max1≤j≤n(Sj − jµ) was precisely the question that were interested Erd¨osand Kac (1946) in. They found

r Z x √ 2 −y 2/2 lim P( max (Sj − jµ) ≤ nx) = e dy n→∞ 1≤j≤n π 0

The limit behavior of Mn = max1≤j≤n(Sj − jµ) was precisely the question that were interested Erd¨osand Kac (1946) in. They found

r Z x √ 2 −y 2/2 lim P( max (Sj − jµ) ≤ nx) = e dy n→∞ 1≤j≤n π 0

for all x ≥ 0, provided X1, X2,... are iid with mean µ and var(X1) = 1. Erdösand Kac described their method of proof as follows: ,,The proof of all these theorems follow the same pattern. It is first proved that the limiting distribution exists and is independent of the distribution of the Xi ’s; then the distribution of Xi ’s is chosen conveniently so that the limiting distribution can be calculated explicitly.” Perhaps the story ,,invariance principle” have realy started with their providence. 1Paul Erdös(26 March 1913 – 20 September 1996) was a Jewish-Hungarian mathematician 2Mark Kac (3 August 1914 – 26 October 1984) was a Polish American mathematician (Rouen 2015) Some functional limit theorems 13 / 94 It is worth to note that Erdösand Kac worked out a number of other interesting functionals h((Sk , k = 1,..., n)). For example, they established arcsin law

n −1 X 2 √ lim P(n 1S >0 ≤ x) = arcsin( x), 0 ≤ x ≤ 1 n→∞ k π k=1

−2 Pn 2 and considered also limit behavior of functionals n k=1 Sk and −3/2 Pn n k=1 |Sk |.

(Rouen 2015) Some functional limit theorems 14 / 94 It is worth to note that Erd¨osand Kac worked out a number of other interesting functionals h((Sk , k = 1,..., n)). For example, they established arcsin law

n −1 X 2 √ lim P(n 1S >0 ≤ x) = arcsin( x), 0 ≤ x ≤ 1 n→∞ k π k=1

−2 Pn 2 and considered also limit behavior of functionals n k=1 Sk and −3/2 Pn n k=1 |Sk |.

(Rouen 2015) Some functional limit theorems 14 / 94 Consider the stochastic process

ζn(t) = Sbntc+(nt−bntc)Xbntc+1,

t ∈ [0, 1], where bac denotes the integer part of the number a.

Donsker’s3 contribution

Donsker (1951) provided a full generalization of Erd¨osand Kac technique by providing explicit embeddings of the sequance Sk , k = 1,..., n into a continuous time stochastic process ζn(t), t ∈ [0, 1] and by establishing the limiting distribution of a −1/2 general continuous functional h(n ζn).

3Monroe David Donsker (October 17, 1924 – June 8, 1991) was an American mathematician and a professor of mathematics at New York University. (Rouen 2015) Some functional limit theorems 15 / 94 Donsker’s3 contribution

Consider the stochastic process

ζn(t) = Sbntc+(nt−bntc)Xbntc+1,

t ∈ [0, 1], where bac denotes the integer part of the number a.

Theorem [Donsker’s IP (sometimes Donsker-Prohorov IP)] 2 2 If (Xi ) are i.i.d. zero mean random variables and EXi = σ > 0, then

−1/2 −1 D n σ ζn −−−→ W n→∞

in the space C[0, 1], where W = (Wt , t ∈ [0, 1]) is a standard Wiener process.

Donsker (1951) established the ﬁrst universal functional limit theorem, called by him an invariance principle. To be more precise let us remind, that C[0, 1] is a Banach space of continuous functions x : [0, 1] → R endowed with the sup-norm

||x|| = sup |x(t)|, x ∈ C[0, 1]. t∈[0,1]

(Rouen 2015) Some functional limit theorems 16 / 94 Donsker (1951) established the ﬁrst universal functional limit theorem, called by him an invariance principle. To be more precise let us remind, that C[0, 1] is a Banach space of continuous functions x : [0, 1] → R endowed with the sup-norm

||x|| = sup |x(t)|, x ∈ C[0, 1]. t∈[0,1]

BackLIP

Theorem [Donsker’s IP (sometimes Donsker-Prohorov IP)] 2 2 If (Xi ) are i.i.d. zero mean random variables and EXi = σ > 0, then

−1/2 −1 D n σ ζn −−−→ W n→∞ in the space C[0, 1], where W = (Wt , t ∈ [0, 1]) is a standard Wiener process.

(Rouen 2015) Some functional limit theorems 16 / 94 This result means particularly, that for any continuous function f : C[0, 1] → R we have

−1/2 −1 lim P(f (n σ ζn) ≤ x) = P(f (W ) ≤ x) n→∞ for all continuity points x of the limit distribution. The set of interesting examples of functions f includes

f (x) = sup x(t), f (x) = sup |x(s) − x(t)|, 0≤t≤1 0<|t−s|≤δ Z 1 Z 1 f (x) = x2(t) dt, f (x) |x(t)| dt 0 0 and many others.

(Rouen 2015) Some functional limit theorems 17 / 94 Taking f (x) = max0≤t≤1[x(t) − tx(1)] we deduce from Donsker’s IP theorem (assuming σ2 = 1), for each a ∈ R,

√ −2a2 lim P( max (Sj − jX n) > na) = P( max [Wt − tW1] > a) = e . n→∞ 1≤j≤n 0≤t≤1

Going back to Example 2 we ﬁnd that an approximate rejection region ∗ ∗ 0 for H0 : k = 0 against H1 : k > 1 and µ > µ is given by q −1/2 −1/2 n max (Sj − jX n) ≥ log(α ) 1≤j≤n and has an asymptotic level α.

(Rouen 2015) Some functional limit theorems 18 / 94 Theorem (self-normalized version (b) of Donsker’s IP)

If (Xi ) are i.i.d. random variables. Then the convergence

ζn D q −−−→ W 2 2 n→∞ X1 + ··· + Xn

in the space C[0, 1] holds if and only if EX1 = 0 and X1 ∈ DNA.

2 2 In the case where σ = var(X1) is unknown one can use its estimator σb as in the CLT. This leads to the following self-normalized extensions of Donsker’s result. Theorem (self-normalized version (a) of Donsker’s IP) 2 2 If (Xi ) are i.i.d. zero mean random variables and σ = EXi < ∞ then

1 D √ ζn −−−→ W nσb n→∞ in the space C[0, 1].

(Rouen 2015) Some functional limit theorems 19 / 94 2 2 In the case where σ = var(X1) is unknown one can use its estimator σb as in the CLT. This leads to the following self-normalized extensions of Donsker’s result. Theorem (self-normalized version (a) of Donsker’s IP) 2 2 If (Xi ) are i.i.d. zero mean random variables and σ = EXi < ∞ then

1 D √ ζn −−−→ W nσb n→∞ in the space C[0, 1].

Theorem (self-normalized version (b) of Donsker’s IP)

If (Xi ) are i.i.d. random variables. Then the convergence

ζn D q −−−→ W 2 2 n→∞ X1 + ··· + Xn in the space C[0, 1] holds if and only if EX1 = 0 and X1 ∈ DNA.

(Rouen 2015) Some functional limit theorems 19 / 94 Latter Prokhorov (1956) completed the general theory of weak convergence of measures in metric spaces and gave general conditions for convergence of continuous stochastic processes.

Before we pass to an other example let us shortly look at the follow-up of the ,,invariance principle” story. After Donsker’s paper Kolmogorov4 and Prokhorov5 (1954) connected the invariance principle with theorems about weak convergence of measures.

4Andrey Nikolaevich Kolmogorov (25 April 1903 20 October 1987) was a 20th-century Soviet mathematician who made signiﬁcant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. 5Yuri Vasilyevich Prokhorov (15 December 1929 16 July 2013) was a Russian mathematician, active in the ﬁeld of probability theory. (Rouen 2015) Some functional limit theorems 20 / 94 Before we pass to an other example let us shortly look at the follow-up of the ,,invariance principle” story. After Donsker’s paper Kolmogorov4 and Prokhorov5 (1954) connected the invariance principle with theorems about weak convergence of measures. Latter Prokhorov (1956) completed the general theory of weak convergence of measures in metric spaces and gave general conditions for convergence of continuous stochastic processes.

4Andrey Nikolaevich Kolmogorov (25 April 1903 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. 5Yuri Vasilyevich Prokhorov (15 December 1929 16 July 2013) was a Russian mathematician, active in the field of probability theory. (Rouen 2015) Some functional limit theorems 20 / 94 ,,The ”invariance principles” of probability theory are mathematically of (n) the following form: a sequence of stochastic processes {xt } induces a sequence of measures µn on some suitable topological function space S, and one proves that the measures converge weakly to a measure µ corresponding to a limiting process {xt }. Weak convergence of measures means here that Z Z lim fdµn = fdµ (wc definition (a)) n→∞ S S for all bounded continuous real-valued functions f on S.

Lamperti’s 6contribution

Next Lamperti explained his viewpoint on the story: John Lamperti. On convergence of stochastic processes. Trans. Amer. Math. Soc. 104, 430–435 (1962).

6Professor (emeritus), Department of Mathematics, Dartmouth College (Rouen 2015) Some functional limit theorems 21 / 94 Lamperti’s 6contribution

Next Lamperti explained his viewpoint on the story: John Lamperti. On convergence of stochastic processes. Trans. Amer. Math. Soc. 104, 430–435 (1962).

,,The ”invariance principles” of probability theory are mathematically of (n) the following form: a sequence of stochastic processes {xt } induces a sequence of measures µn on some suitable topological function space S, and one proves that the measures converge weakly to a measure µ corresponding to a limiting process {xt }. Weak convergence of measures means here that Z Z lim fdµn = fdµ (wc deﬁnition (a)) n→∞ S S for all bounded continuous real-valued functions f on S.

6Professor (emeritus), Department of Mathematics, Dartmouth College (Rouen 2015) Some functional limit theorems 21 / 94 This is equivalent to the condition that

(n) lim L(f (xt )) = L(f (xt ))), (wc deﬁnition (b)) n→∞ where L(X ) is the distribution function of the random variable X , f () is a real-valued function S-continuous almost everywhere (µ), and the limit is in the sense of the usual weak convergence of distributions. Equation (wc deﬁnition (b)) is usually the real center of interest, for many ,,limit-distribution theorems” are implicit in it.” Lamperti also points:

,,It is clear that for given (µn) and µ, the better theorem of this kind would be the one in which (wc deﬁnition (b)) is proved for the larger class of functions f .”

(Rouen 2015) Some functional limit theorems 22 / 94 One can think about say log-returns of certain ﬁnancial instrument.

Third example (number of big claims)

Let Y1, Y2,..., Yn be a sequence of independent and identically distributed random variables with cumulative distribution function F

and denote by Xi (u) = 1{Yi ≥u} the indicator variable. The quantity u ∈ (0, ∞) is a ﬁxed threshold which is exceeded by Yi with probability

p = P(Xi (u) = 1) = E(Xi (u)) = P(Yi > u) = F (u),

where F denotes the tail probability of Yi .

(Rouen 2015) Some functional limit theorems 23 / 94 Third example (number of big claims)

Let Y1, Y2,..., Yn be a sequence of independent and identically distributed random variables with cumulative distribution function F

and denote by Xi (u) = 1{Yi ≥u} the indicator variable. The quantity u ∈ (0, ∞) is a ﬁxed threshold which is exceeded by Yi with probability

p = P(Xi (u) = 1) = E(Xi (u)) = P(Yi > u) = F (u),

where F denotes the tail probability of Yi .

One can think about say log-returns of certain ﬁnancial instrument.

(Rouen 2015) Some functional limit theorems 23 / 94 Let us consider all the moving windows of length k in the sequence Y1, Y2,..., Yn, namely,

Yi , Yi+1,..., Yi+k−1, i = 1, 2,..., n − k + 1.

Deﬁne the k-scan exceeds process as follows:

i+k−1 i+k−1 (i) X X Sk (u) = Xj (u) = 1(u,∞)(Yi ), i = 1,..., n − k + 1. j=i j=i

This process counts the number of random variables, among Yi , Yi+1,..., Yi+k−1, whose value exceeds threshold u, while

(i) Sn,k (u) = max S (u) 1≤i≤n−k+1 k expresses the maximum number of exceeds occurrence among all possible moving windows of length k, in the sequence Y1, Y2,..., Yn.

(Rouen 2015) Some functional limit theorems 24 / 94 Asymptotic distributions of

Sn,k (u) and max q(k)Sn,k (u) n 1≤k≤n where q(k) is a windows lengths of interest are of great importance. To understand asymptotic behaviour of these quantities one needs to consider polygonal line processes on function spaces with stronger topology as that of the space of continuous functions.

(Rouen 2015) Some functional limit theorems 25 / 94 Asymptotic distributions of

(Rouen 2015) Some functional limit theorems 25 / 94 Hölderspaces The introduction of Hölder7 spaces answers the purpose to quantify the global smoothness of functions by controlling their modulus of uniform continuity. We shall discuss 1 definition of Hölderspaces; 2 some geometry of this space, including characterization of compact sets; 3 weak convergence of probability measures on these spaces; 4 conditions for random processes to have versions with Hölderianpaths;

5 general conditions of tightness of measures induced by polygonal line processes.

7 Otto Ludwig Hölder(December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart. Noted for many results and ideas including: Hölder’s inequality, the Jordan-Hölder theorem, Höldercondition (or Höldercontinuity) which are used in many areas of analysis and stochastic processes.

(Rouen 2015) Some functional limit theorems 26 / 94 The set Hα is a Banach space when endowed with the norm

kxkα = |x(0)| + ωα(x, 1), x ∈ Hα.

However neither of Hα is separable. [To see this it is enough to consider the set of functions {fs : 0 ≤ s ≤ 1} α where for each s, fs (t) = |t − s| , 0 ≤ t ≤ 1, and observe that the distance 0 kfs − fs0 kα ≥ 2 for s 6= s .] If 0 ≤ α < β ≤ 1, Hβ is topologically embedded in Hα and all these H¨olderspaces are topologically embedded in the classical Banach space C[0, 1].

o The spaces Hα

For ﬁxed 0 ≤ α ≤ 1, denote by Hα := Hα[0, 1] the set of real-valued continuous functions x : [0, 1] → R such that ωα(x, 1) < ∞, where |x(t) − x(s)| ωα(x, δ) := sup α , 0 ≤ δ < 1 0<|t−s|<δ |t − s| is so called H¨olderianmodulus of continuity of the function x.

(Rouen 2015) Some functional limit theorems 27 / 94 However neither of Hα is separable. [To see this it is enough to consider the set of functions {fs : 0 ≤ s ≤ 1} α where for each s, fs (t) = |t − s| , 0 ≤ t ≤ 1, and observe that the distance 0 kfs − fs0 kα ≥ 2 for s 6= s .] If 0 ≤ α < β ≤ 1, Hβ is topologically embedded in Hα and all these H¨olderspaces are topologically embedded in the classical Banach space C[0, 1].

o The spaces Hα

kxkα = |x(0)| + ωα(x, 1), x ∈ Hα.

(Rouen 2015) Some functional limit theorems 27 / 94 If 0 ≤ α < β ≤ 1, Hβ is topologically embedded in Hα and all these H¨olderspaces are topologically embedded in the classical Banach space C[0, 1].

o The spaces Hα

kxkα = |x(0)| + ωα(x, 1), x ∈ Hα.

(Rouen 2015) Some functional limit theorems 27 / 94 o The spaces Hα

kxkα = |x(0)| + ωα(x, 1), x ∈ Hα.

(Rouen 2015) Some functional limit theorems 27 / 94 o For any 0 ≤ α < 1, the subspace Hα is closed and separable. [The closeness follows straightforward. The separability follows from the o existence of Schauder basis in each Hα (to be stated below).] o Hence, for any 0 ≤ α < 1, the space Hα is a separable Banach space with the norm k · kα,

|x(t) − x(s)| o kxkα = |x(0)| + sup α , x ∈ Hα. 0<|t−s|<1 |t − s|

o H0 ' C[0, 1].

To remedy the non separability drawback of Hα, one introduces its o subspace Hα:

o Hα = {x ∈ Hα : lim ωα(x, δ) = 0}. δ→0

(Rouen 2015) Some functional limit theorems 28 / 94 o Hence, for any 0 ≤ α < 1, the space Hα is a separable Banach space with the norm k · kα,

|x(t) − x(s)| o kxkα = |x(0)| + sup α , x ∈ Hα. 0<|t−s|<1 |t − s|

o H0 ' C[0, 1].

To remedy the non separability drawback of Hα, one introduces its o subspace Hα:

o Hα = {x ∈ Hα : lim ωα(x, δ) = 0}. δ→0

(Rouen 2015) Some functional limit theorems 28 / 94 o H0 ' C[0, 1].

To remedy the non separability drawback of Hα, one introduces its o subspace Hα:

o Hα = {x ∈ Hα : lim ωα(x, δ) = 0}. δ→0

o For any 0 ≤ α < 1, the subspace Hα is closed and separable. [The closeness follows straightforward. The separability follows from the o existence of Schauder basis in each Hα (to be stated below).] o Hence, for any 0 ≤ α < 1, the space Hα is a separable Banach space with the norm k · kα,

|x(t) − x(s)| o kxkα = |x(0)| + sup α , x ∈ Hα. 0<|t−s|<1 |t − s|

(Rouen 2015) Some functional limit theorems 28 / 94 To remedy the non separability drawback of Hα, one introduces its o subspace Hα:

o Hα = {x ∈ Hα : lim ωα(x, δ) = 0}. δ→0

|x(t) − x(s)| o kxkα = |x(0)| + sup α , x ∈ Hα. 0<|t−s|<1 |t − s|

o H0 ' C[0, 1].

(Rouen 2015) Some functional limit theorems 28 / 94 Let us denote by Dj the set of dyadic numbers in [0, 1] of level j, i.e. −j j−1 D0 = {0, 1}, Dj = (2l − 1)2 ; 1 ≤ l ≤ 2 , j ≥ 1.

Write for r ∈ Dj , j ≥ 0, r − := r − 2−j , r + := r + 2−j .

For r ∈ Dj , j ≥ 1, the triangular Faber-Schauder functions Λr are continuous, piecewise aﬃne with support [r −, r +] and taking the value 1 at r:  2j (t − r −) if t ∈ (r −, r];  j + + Λr (t) = 2 (r − t) if t ∈ (r, r ];  0 else.

o Schauder basis in Hα o One of the most interesting features of the spaces Hα is the existence of a basis of triangular functions8.

8Ciesielski (1960) (Rouen 2015) Some functional limit theorems 29 / 94 For r ∈ Dj , j ≥ 1, the triangular Faber-Schauder functions Λr are continuous, piecewise aﬃne with support [r −, r +] and taking the value 1 at r:  2j (t − r −) if t ∈ (r −, r];  j + + Λr (t) = 2 (r − t) if t ∈ (r, r ];  0 else.

o Schauder basis in Hα o One of the most interesting features of the spaces Hα is the existence of a basis of triangular functions8.

Let us denote by Dj the set of dyadic numbers in [0, 1] of level j, i.e. −j j−1 D0 = {0, 1}, Dj = (2l − 1)2 ; 1 ≤ l ≤ 2 , j ≥ 1.

Write for r ∈ Dj , j ≥ 0, r − := r − 2−j , r + := r + 2−j .

8Ciesielski (1960) (Rouen 2015) Some functional limit theorems 29 / 94 o Schauder basis in Hα o One of the most interesting features of the spaces Hα is the existence of a basis of triangular functions8.

Let us denote by Dj the set of dyadic numbers in [0, 1] of level j, i.e. −j j−1 D0 = {0, 1}, Dj = (2l − 1)2 ; 1 ≤ l ≤ 2 , j ≥ 1.

Write for r ∈ Dj , j ≥ 0, r − := r − 2−j , r + := r + 2−j .

8Ciesielski (1960) (Rouen 2015) Some functional limit theorems 29 / 94 Λr (t) 6 1 £C £ C £ C £ C £ C £ C £ C £ C £ C - − + 0 r r -r t 2−j

Figure: The Faber-Schauder triangular function Λr

When j = 0, Λ0(t) = 1 − t, Λ1(t) = t, t ∈ [0, 1].

(Rouen 2015) Some functional limit theorems 30 / 94 o Theorem (existence of Schauder basis in Hα)

For each 0 ≤ α < 1, the sequence {Λr ; r ∈ Dj , j ≥ 0} is a Schauder basis o o in Hα: each x ∈ Hα has a unique expansion

∞ X X x(t) = λr (x)Λr (t), t ∈ [0, 1],

j=0 r∈Dj where the Schauder scalar coeﬃcients λr (x) are given by

x(r +) + x(r −) λ (x) = x(r) − , r ∈ D , j ≥ 1, r 2 j and in the special case j = 0 by

λ0(x) = x(0), λ1(x) = x(1).

(Rouen 2015) Some functional limit theorems 31 / 94 Proof. o Let x ∈ Hα. The partial sum

J X X EJ (x)(t) := λr (x)Λr (t), t ∈ [0, 1],

j=0 r∈Dj gives the linear interpolation of x by a polygonal line between the dyadic points of level at most J. One can check that

−J kx − EJ xkα ≤ 4ωα(x, 2 ).

This estimate proves the result.

(Rouen 2015) Some functional limit theorems 32 / 94 Equivalent norm

o Theorem (equivalent sequential norm in Hα)[Ciesielski (1960)]

The norm kxkα is equivalent to the following sequence norm :

seq jα kxkα := sup 2 max |λr (x)|, j≥0 r∈Dj

that is, there exist two positive constants c, C > 0 such that

seq seq ckxkα ≤ kxkα ≤ Ckxkα

o for any x ∈ Hα.

(Rouen 2015) Some functional limit theorems 33 / 94 Proof. It is enough to follow along the lines of the proof of Arzela-Ascoli theorem.

Compact sets

o Next we characterize compact sets in Hα.

o Theorem (compact sets in Hα) o Let 0 ≤ α < 1. The set K ⊂ Hα is compact if and only if (i) K is bounded: sup ||x|| < ∞; x∈K (ii) K is α-equicontinuous:

lim sup ωα(x, δ) = 0. δ→0 x∈K

(Rouen 2015) Some functional limit theorems 34 / 94 Compact sets

o Next we characterize compact sets in Hα.

o Theorem (compact sets in Hα) o Let 0 ≤ α < 1. The set K ⊂ Hα is compact if and only if (i) K is bounded: sup ||x|| < ∞; x∈K (ii) K is α-equicontinuous:

lim sup ωα(x, δ) = 0. δ→0 x∈K

Proof. It is enough to follow along the lines of the proof of Arzela-Ascoli theorem.

(Rouen 2015) Some functional limit theorems 34 / 94 We associate to ρ the H¨olderspace

o Hρ := {x ∈ C[0, 1] : lim ωρ(x, δ) = 0}, δ→0 equiped with the norm

kxkρ := |x(0)| + ωρ(x, 1).

Generalized H¨olderspaces

Let ρ be a real valued non decreasing function on [0, 1], null and right continuous at 0, positive on (0, 1]. For a function x : [0, 1] → R put

|x(t) − x(s)| ωρ(x, δ) := sup . s,t∈[0,1], ρ(t − s) 0

(Rouen 2015) Some functional limit theorems 35 / 94 Generalized H¨olderspaces

Let ρ be a real valued non decreasing function on [0, 1], null and right continuous at 0, positive on (0, 1]. For a function x : [0, 1] → R put

|x(t) − x(s)| ωρ(x, δ) := sup . s,t∈[0,1], ρ(t − s) 0

We associate to ρ the H¨olderspace

o Hρ := {x ∈ C[0, 1] : lim ωρ(x, δ) = 0}, δ→0 equiped with the norm

kxkρ := |x(0)| + ωρ(x, 1).

(Rouen 2015) Some functional limit theorems 35 / 94 o Under some restrictions on ρ ( denoted ρ ∈ R), the space Hρ is a separable Banach space and its norm is equivalent to sequential one:

seq 1 o kxkρ := sup −j max |λj,t (x)|, x ∈ Hρ. j≥0 ρ(2 ) t∈Dj

Particularly the function ρ = ρα,β, 0 < α < 1, β ∈ R, deﬁned by:

α β ρα,β(h) := h ln (c/h), 0 < h ≤ 1, for a suitable constant c belongs to R. o o o o We write Hα,β for Hρ when ρ = ρα,β and we abbreviate Hα,0 in Hα.

(Rouen 2015) Some functional limit theorems 36 / 94 o Compact sets in Hρ

Theorem o Let ρ ∈ R. The set K ⊂ Hρ is compact if and only if (i) K is bounded: sup ||x|| < ∞; x∈K (ii) K is ρ-equicontinuous:

lim sup ωρ(x, δ) = 0. δ→0 x∈K

(Rouen 2015) Some functional limit theorems 37 / 94 The first result in this direction goes back to the Kolmogorov’s sufficient condition for the existence of a continuous version of ξ. In fact, the same condition gives a version of ξ with some α-Hölder regularity.

Theorem [Kolmogorov’s, on the existence of a continuous version]

Assume that the process ξ = (ξt , t ∈ [0, 1]} satisﬁes the condition: there exists δ > 0 and γ > 1 such that for each t ∈ [0, 1] and h > 0 such that t + h ∈ [0, 1] −γ 1+δ P(|ξt+h − ξt | > λ) ≤ cλ h . o Then ξ admits a version with almost all paths in the space Hα for α < δ/γ.

(Rouen 2015) Some functional limit theorems 38 / 94 In fact, the same condition gives a version of ξ with some α-H¨older regularity.

Theorem [Kolmogorov’s, on the existence of a continuous version]

Random processes with paths in Hölderspaces As random processes induce probability measures on function spaces, it is natural to ask the question, when a random process, say, o ξ = (ξt , t ∈ [0, 1]), induces a probability measure on Hα? The first result in this direction goes back to the Kolmogorov’s sufficient condition for the existence of a continuous version of ξ.

(Rouen 2015) Some functional limit theorems 38 / 94 Theorem [Kolmogorov’s, on the existence of a continuous version]

(Rouen 2015) Some functional limit theorems 38 / 94 Random processes with paths in Hölderspaces As random processes induce probability measures on function spaces, it is natural to ask the question, when a random process, say, o ξ = (ξt , t ∈ [0, 1]), induces a probability measure on Hα? The first result in this direction goes back to the Kolmogorov’s sufficient condition for the existence of a continuous version of ξ. In fact, the same condition gives a version of ξ with some α-Hölder regularity.

Theorem [Kolmogorov’s, on the existence of a continuous version]

(Rouen 2015) Some functional limit theorems 38 / 94 Examples

Wiener process (Brownian motion)

Deﬁnition A standard Wiener process (also called Brownian motion ) is a stochastic process W = (Wt , t ≥ 0) with the following properties:

(1) W0 = 0 (2) the process has stationary and independent increments

(3) the increment Wt+h − Wt ∼ N (0, h)

(4) with probability 1, the function t → Wt is continuous.

(Rouen 2015) Some functional limit theorems 39 / 94 o Hence, W has version with paths in Hα for any α < 1/2 − 1/q. Since o q > 2 is arbitrary we conclude that W has a version with paths in Hα for any α < 1/2. This is the best possible regularity power of Wiener process. o In the class {Hα,β}, the best possible H¨olderspace for a version of o Wiener process is H1/2.β, where β > 1/2.

Consider W = (W (t), t ∈ [0, 1]). Since W (t + h) − W (t) has normal distribution with zero mean and variance h, we ﬁnd

−q q −q q/2 P(|W (t +h)−W (t)| > λ) ≤ λ E|W (t +h)−W (t)| ≤ cqλ |h| .

(Rouen 2015) Some functional limit theorems 40 / 94 Consider W = (W (t), t ∈ [0, 1]). Since W (t + h) − W (t) has normal distribution with zero mean and variance h, we ﬁnd

−q q −q q/2 P(|W (t +h)−W (t)| > λ) ≤ λ E|W (t +h)−W (t)| ≤ cqλ |h| .

o Hence, W has version with paths in Hα for any α < 1/2 − 1/q. Since o q > 2 is arbitrary we conclude that W has a version with paths in Hα for any α < 1/2. This is the best possible regularity power of Wiener process. o In the class {Hα,β}, the best possible H¨olderspace for a version of o Wiener process is H1/2.β, where β > 1/2.

(Rouen 2015) Some functional limit theorems 40 / 94 Fractional Brownian motion

Deﬁnition Standard fractional Brownian motion is a Gaussian process BH = (BH (t), t ∈ [0, 1]) with expectation zero and the following covariance function:

1 2H 2H 2H E[BH (t)BH (s)] = 2 (|t| + |s| − |t − s| ), where H is a real number in (0, 1), called the Hurst index or Hurst exponent associated with the fractional Brownian motion.

(Rouen 2015) Some functional limit theorems 41 / 94 Since W is a Gaussian process and W (t + h) − W (t) has normal distribution with mean zero and variance h2H , we have

−q q P(|WH (t + h) − WH (t)| > λ) ≤ λ E|W (t + h) − W (t)| −q Hq ≤ cqλ |h| .

o Hence, W has a version with paths in Hα for any α < H − 1/q. Since q > 2 is arbitrary we conclude that W has a version with paths o in Hα for any α < H.

Let WH = (WH (t), t ∈ [0, 1]) be a standard fractional Brownian motion with Hurst exponent H, 0 < H < 1.

(Rouen 2015) Some functional limit theorems 42 / 94 o Hence, W has a version with paths in Hα for any α < H − 1/q. Since q > 2 is arbitrary we conclude that W has a version with paths o in Hα for any α < H.

Let WH = (WH (t), t ∈ [0, 1]) be a standard fractional Brownian motion with Hurst exponent H, 0 < H < 1. Since W is a Gaussian process and W (t + h) − W (t) has normal distribution with mean zero and variance h2H , we have

−q q P(|WH (t + h) − WH (t)| > λ) ≤ λ E|W (t + h) − W (t)| −q Hq ≤ cqλ |h| .

(Rouen 2015) Some functional limit theorems 42 / 94 Let WH = (WH (t), t ∈ [0, 1]) be a standard fractional Brownian motion with Hurst exponent H, 0 < H < 1. Since W is a Gaussian process and W (t + h) − W (t) has normal distribution with mean zero and variance h2H , we have

−q q P(|WH (t + h) − WH (t)| > λ) ≤ λ E|W (t + h) − W (t)| −q Hq ≤ cqλ |h| .

o Hence, W has a version with paths in Hα for any α < H − 1/q. Since q > 2 is arbitrary we conclude that W has a version with paths o in Hα for any α < H.

(Rouen 2015) Some functional limit theorems 42 / 94 Deﬁnition of weak convergence of probability measures

A sequence of probability measures (Pn) ⊂ P(S) converges weakly to a w probability measure P ∈ P(S) (denoted Pn −−−→ P) if n→∞ Z Z lim f (x)Pn(dx) = f (x)P(dx) n→∞ S S for each bounded continuous function f : S → R.

Weak convergence and tightness: general

Let (S, d) be a complete separable metric space where S denotes the set and d is the metric.

Let BS denote the σ-algebra of Borel sets associated with the topology induced on S by d.

Let P(S) denote the set of all probability measures on (S, BS ).

(Rouen 2015) Some functional limit theorems 43 / 94 Weak convergence and tightness: general

Let (S, d) be a complete separable metric space where S denotes the set and d is the metric.

Let BS denote the σ-algebra of Borel sets associated with the topology induced on S by d.

Let P(S) denote the set of all probability measures on (S, BS ).

Deﬁnition of weak convergence of probability measures

(Rouen 2015) Some functional limit theorems 43 / 94 There is a metric ρ for P(S) that induces weak convergence, that is, w Pn −−−→ P if and only if limn→∞ ρ(Pn, P) = 0. n→∞ Examples provide Bounded Lipschitz metric n Z Z o

dBL(P, Q) = sup fdP − fdQ : f ∈ L1

|f (x) − f (y)| f ∈ L1 : sup |f (x)| + sup ≤ 1. x x6=y d(x, y) and Prohorov metric

π(P, Q) = inf{ε > 0 : P(A) ≤ Q(Aε) + ε, A ∈ B(S)}, where Aε = {x : d(x, A) ≤ ε}.

(Rouen 2015) Some functional limit theorems 44 / 94 Deﬁnition of tightness of a set of probability measures

A family of probability measures P0 ⊂ P(S) is tight if for each ε > 0 there is a compact set Kε ⊂ S such that

P(Kε) > 1 − ε for all P ∈ P0.

Deﬁnition of relative compactness of a set of probability measures

A family of probability measures P0 ⊂ P(S) is (weakly) relatively compact if each sequence (Pn) ⊂ P0 has a subsequence that converges weakly to a probability measure P ∈ P(S).

(Rouen 2015) Some functional limit theorems 45 / 94 Deﬁnition of relative compactness of a set of probability measures

A family of probability measures P0 ⊂ P(S) is (weakly) relatively compact if each sequence (Pn) ⊂ P0 has a subsequence that converges weakly to a probability measure P ∈ P(S).

Deﬁnition of tightness of a set of probability measures

A family of probability measures P0 ⊂ P(S) is tight if for each ε > 0 there is a compact set Kε ⊂ S such that

P(Kε) > 1 − ε for all P ∈ P0.

(Rouen 2015) Some functional limit theorems 45 / 94 Corollary (mostly used in asymptotic theory)

Suppose that (Pn) is a tight family of probability measures on (S, BS ) and that there is a probability measure P on (S, BS ) such that each weakly convergent subsequence of (Pn) has limit P. Then

w Pn −−−→ P. n→∞

Prohorov’s theorem

A family of probability measures on (S, BS ) is tight if and only if it is (weakly) relatively compact.

(Rouen 2015) Some functional limit theorems 46 / 94 Prohorov’s theorem

A family of probability measures on (S, BS ) is tight if and only if it is (weakly) relatively compact.

Corollary (mostly used in asymptotic theory)

Suppose that (Pn) is a tight family of probability measures on (S, BS ) and that there is a probability measure P on (S, BS ) such that each weakly convergent subsequence of (Pn) has limit P. Then

w Pn −−−→ P. n→∞

(Rouen 2015) Some functional limit theorems 46 / 94 backtorf

o Theorem [tightness of probability measures in Hρ] o o The sequence (Pn) of probability measures on (Hρ, BHρ ) is tight if and only if the following two conditions hold:

(i) lima→∞ supn Pn(x : |x(0)| > a) = 0; (ii) for each positive ε

lim sup Pn(x : ωρ(x, δ) ≥ ε) = 0. δ→0 n

Weak convergence and tightness in H

o o By BHρ we denote the Borel σ-algebra of subsets of Hρ. o Since Hα is separable and complete it follows by Prohorov’s theorem that relative compactness of a family of probability measures on o o (Hρ, BHρ ) is equivalent to its tightness.

(Rouen 2015) Some functional limit theorems 47 / 94 Weak convergence and tightness in H

backtorf

o Theorem [tightness of probability measures in Hρ] o o The sequence (Pn) of probability measures on (Hρ, BHρ ) is tight if and only if the following two conditions hold:

(i) lima→∞ supn Pn(x : |x(0)| > a) = 0; (ii) for each positive ε

lim sup Pn(x : ωρ(x, δ) ≥ ε) = 0. δ→0 n

(Rouen 2015) Some functional limit theorems 47 / 94 o Tightness of random processes in Hρ

o Suppose now that (Zn) is a sequence of random functions in Hρ. The sequence (Zn) is by deﬁnition tight when the sequence of corresponding

distributions (PZn ) is tight.

o Theorem [Tightness of random functions in Hα]

The sequence (Zn) of random processes with paths in the H¨olderspace o Hα is tight if and only if the following two conditions are satisﬁed:

(i) limb→∞ supn P(|Zn(0)| > b) = 0; (ii) for each ε > 0 lim sup P(ωρ(Zn, δ) ≥ ε) = 0. δ→0 n

(Rouen 2015) Some functional limit theorems 48 / 94 This theorem can be recast by using the equivalent norm.

Theorem [Tightness of random functions using sequential norm] o The sequence (Zn)n≥1 of random elements in Hρ is tight if and only if the two following conditions are fulﬁlled:

(i) limb→∞ supn P(|Zn(0)| > b) = 0; (TCond. 1) (ii) For each positive ε, ! 1 lim sup Pr sup −j max kλr (Zn)k > ε = 0. (TCond. 2) J→∞ n≥1 j>J ρ(2 ) r∈Dj

proof:1 proof:2

(Rouen 2015) Some functional limit theorems 49 / 94 Polygonal line process

Let X1, X2,... be random variables on some probability space (Ω, F, P). For the present, they need not to be independent or stationary. We deﬁne

S0 = 0, Sk = X1 + ··· + Xk , k ≥ 1.

Next we construct from the partial sums S0, S1,..., Sn the polygonal line process ζn = (ζn(t), t ∈ [0, 1]):

ζ (t) = S + (nt − [nt])X , t ∈ [0, 1]. §n [nt] [nt]+1 ¤

Note that ζn(k¦/n) = Sk for k = 0, 1,..., n. ¥

backto:1 backto:2 backto:3

(Rouen 2015) Some functional limit theorems 50 / 94 Lemma [H¨oldernorm of polygonal line function] Let ρ : [0, 1] → R be a weight function satisfying the following properties. 1 ρ is concave. 2 ρ(0) = 0 and ρ is positive on (0, 1]. 3 ρ is non decreasing on [0, 1]. 4 ρ(1) ≥ 1.

Let t0 = 0 < t1 < ··· < tn = 1 be a partition of [0, 1] and f be a real valued polygonal line function on [0, 1] with vertices at the ti ’s, i.e. f is continuous on [0, 1] and its restriction to each interval [ti , ti+1] is an aﬃne function. Deﬁne |f (t) − f (s)| R(s, t) := , 0 ≤ s < t ≤ 1. ρ(t − s)

Then sup R(s, t) = max R(ti , tj ). (ρ-norm) 0≤s

(Rouen 2015) Some functional limit theorems 51 / 94 Proof

Obviously (ρ-norm) will be established if we prove that

R(s, t) ≤ max R(ti , tj ), (ρ-norm bound) 0≤i

(Rouen 2015) Some functional limit theorems 52 / 94 In the ﬁrst conﬁguration,

f (b) − f (a) f (t) − f (s) = (t − s), b − a whence t − s ρ(b − a) R(s, t) = R(a, b) . ρ(t − s) b − a By concavity of ρ, ρ(δh) ≥ δρ(h), for 0 ≤ δ ≤ 1. Put h0 = δh, hence h > h0 and δ = h0/h < 1. Then

h0 ρ(h0) ρ(h) h h0 ρ(h0) ≥ ρ(h) ⇒ ≥ ⇒ ≥ , h > h0. h h0 h ρ(h) ρ(h0)

We obtained that function h 7→ h/ρ(h) is increasing, and so t−s ρ(b−a) ρ(t−s) b−a < 1. This gives R(s, t) ≤ R(a, b).

(Rouen 2015) Some functional limit theorems 53 / 94 In the second conﬁguration, let us parametrize the segment [a, b] by putting t = (1 − u)a + ub, u ∈ [0, 1]. Then t − s = (1 − u)(a − s) + u(b − s) and as f (t) − f (s) is aﬃne on [a, b], f (t) − f (s) = (1 − u)(f (a) − f (s)) + u(f (b) − f (s)). Now to estimate R(s, t), using triangular inequality for the numerator and the concavity of ρ for the denominator gives:

(1 − u)|f (a) − f (s)| + u|f (b) − f (s)| Au + B B0 R(s, t) ≤ = = A0+ , (1 − u)ρ(a − s) + uρ(b − s) Cu + D Cu + D where the constants A, A0,..., D depend on f , ρ, a, b and s (which is ﬁxed here). As ρ is non decreasing, (1 − u)ρ(a − s) + uρ(b − s) ≥ ρ(a − s) > 0, so Cu + D remains positive when u varies between 0 and 1. It follows that the homographic function A0 + B0/(Cu + D) is monotonic on [0, 1] and hence reaches its maximum at u = 0 or at u = 1. This gives R(s, t) ≤ max R(s, a), R(s, b). The bound for R(s, t) in the third conﬁguration is obtained in a completely similar way, so we omit the details.

(Rouen 2015) Some functional limit theorems 54 / 94 Theorem [tightness of polygonal line processes]

Let (ζn) be the polygonal line processes built on (Xk )k≥0. Let ρ ∈ R. −1 o Then (bn ζn)n≥1 is tight in the space Hρ if: 1 (1) max |Xi | converges in probability to 0; ρ(1/n)bn 1≤i≤n (2) for every positive ε,

n 1 o lim lim sup Pr max −j max S[nr +] − S[nr] ≥ bnε = 0. J→∞ n→∞ J≤j≤log n ρ(2 ) r∈Dj

If the Xi ’s have identical distributions, then Condition (2) can be replaced by (2’) for every ε > 0 nPr |X1| ≥ εbnρ(1/n) −−−→ 0. n→∞

popro:2

(Rouen 2015) Some functional limit theorems 55 / 94 Proof

Since ζn(0) = 0 we have to check condition (ii) of Theorem tight:seqnorm only.

Denote by P0 = P0(J, n) the probability appearing in its condition (ii). Then P0 is bounded by P1 + P2 where

n 1 + o P1 := Pr max −j max |ζn(r ) − ζn(r)| ≥ bnε J≤j≤log n ρ(2 ) r∈Dj

and

n 1 + o P2 := Pr sup −j max |ζn(r ) − ζn(r)| ≥ bnε . j>log n ρ(2 ) r∈Dj

(Rouen 2015) Some functional limit theorems 56 / 94 Estimation of P2.

As j > log n, r + − r = 2−j < 1/n and then with r in say [l/n, (l + 1)/n), either r + is in (l/n, (l + 1)/n] or belongs to (l + 1)/n, (l + 2)/n, where 1 ≤ l ≤ n − 2. + In the ﬁrst case, computing ζn(r ) − ζn(r) by linear interpolation of ζn between ζn(l/n) and ζn((l + 1)/n), we obtain

+ −j −j |ζn(r ) − ζn(r| = n|Xl+1|2 ≤ 2 n max |Xi |. 1≤i≤n

If r and r + are in consecutive intervals, then

+ + |ζn(r ) − ζn(r)| ≤ |ζn(r) − ζn((l + 1)/n)| + |ζn((l + 1)/n) − ζn(r )| + ≤ (l + 1)/n − r + r − (l + 1)/n n max |Xi | 1≤i≤n −j = 2 n max |Xi |. 1≤i≤n

(Rouen 2015) Some functional limit theorems 57 / 94 + This estimate of |ζn(r ) − ζn(r)| leads to

n 1 −j o P2 ≤ Pr sup −j n2 max |Xi | ≥ bnε j>log n ρ(2 ) 1≤i≤n n o = Pr max |Xi | ≥ ρ(1/n)bnε , 1≤i≤n for n large enough, whence by Condition (1), limn→∞ P2 = 0.

(Rouen 2015) Some functional limit theorems 58 / 94 Estimation of P1.

Let uk = [nr]. Then uk ≤ nr ≤ 1 + uk and + 1 + uk ≤ uk+1 ≤ nr ≤ 1 + uk+1. Therefore + + |ζn(r ) − ζn(r)| ≤ |ζn(r ) − Suk+1 | + |Suk+1 − Suk | + |Suk − ζn(r)|. + Since |Suk − ζn(r)| ≤ |X1+uk | and |ζn(r ) − Suk+1 | ≤ |X1+uk+1 | we obtain P1 ≤ P1,1 + 2P1,2, where n 1 εo P1,1 := Pr max −j max |Suk+1 − Suk | ≥ bn J≤j≤log n ρ(2 ) r∈Dj 2 n 1 εo P1,2 := Pr max max |Xi | ≥ bn . J≤j≤log n ρ(2−j ) 1≤i≤n 4

In P1,2, the maximum over j is realized for j = [log n], so limn→∞ P1,2 = 0 by Condition (1).

By Condition 2 we have limJ→∞ lim supn→∞ P1,1 = 0. Gathering now all the estimates, we complete the proof.

(Rouen 2015) Some functional limit theorems 59 / 94 Theorem [tightness via moments behaviour]

Let ζn be the polygonal line process built on stationary sequence (Xk )k≥0. Let 0 < H < 1 and 0 ≤ α < H. Assume that the following condition is satisﬁed: for some q > 1/(H − α) there is a ﬁnite constant cq > 0 such that for each m ≥ 1 q qH E|Sm| ≤ cqm . −H o Then thes sequence (n ζn) is tight in any Hα with 0 ≤ α < H − 1/q.

BackLIP

(Rouen 2015) Some functional limit theorems 60 / 94 Proof

One has to check the condition (ii) of Theorem tight:mom only. Fix ε > 0 and J ≥ 1. Then

n αj H o Pr max 2 max S[nr +] − S[nr] ≥ n ε ≤ J≤j≤log n r∈Dj log n log n −q −qH X qαj j q −q X −qHj+qαj+j ε n 2 2 max E S[nr +] − S[nr] ≤ ε 2 r∈Dj j=J j=J by stationarity and condition (ii). This ends the proof.

(Rouen 2015) Some functional limit theorems 61 / 94 Proof.

By Donsker invariance principle DonskerIP ﬁnite dimensional distributions of −1/2 −1 n σ ζn converge to those of W . Hence one needs to prove tightness −1/2 o tightness via moments behaviour of (n ζn) in Hα. To this aim we use Theorem with H = 1/2.

Lamperti’s invariance principle

Theorem [Lamperti’s IP]

Let ζn be the partial sums process built on iid zero mean random variables q 2 2 (Xk )k≥0. Fix q > 2. Assume E|X1| < ∞ and let σ = EX1 . Then

−1/2 −1 D n σ ζn −−−→ W n→∞

o in any Hα with 0 ≤ α < 1/2 − 1/q.

(Rouen 2015) Some functional limit theorems 62 / 94 Lamperti’s invariance principle

Theorem [Lamperti’s IP]

Let ζn be the partial sums process built on iid zero mean random variables q 2 2 (Xk )k≥0. Fix q > 2. Assume E|X1| < ∞ and let σ = EX1 . Then

−1/2 −1 D n σ ζn −−−→ W n→∞

o in any Hα with 0 ≤ α < 1/2 − 1/q.

Proof.

(Rouen 2015) Some functional limit theorems 62 / 94 By Rosenthal’s inequality9 we deduce

m m q/2 q h X 2 X qi q q/2 E|Sm| ≤ cq E|Xk | + E|Xk | ≤ cqE|X1| m k=1 k=1 where the constant cq depends on q only.

9 If Y1, Y2,... are independent mean zero random variables then for any q ≥ 2 there is a constant cq such that for any m ≥ 1

m m m p h q/2 i X X 2 X q E Yk ≤ cq E|Yk | + E|Yk | k=1 k=1 k=1

(Rouen 2015) Some functional limit theorems 63 / 94 By Rosenthal’s inequality9 we deduce

m m q/2 q h X 2 X qi q q/2 E|Sm| ≤ cq E|Xk | + E|Xk | ≤ cqE|X1| m k=1 k=1 where the constant cq depends on q only.

9 If Y1, Y2,... are independent mean zero random variables then for any q ≥ 2 there is a constant cq such that for any m ≥ 1

m m m p h q/2 i X X 2 X q E Yk ≤ cq E|Yk | + E|Yk | k=1 k=1 k=1

(Rouen 2015) Some functional limit theorems 63 / 94 Theorem [NSC for Lamperti IP]

Let ζn be the partial sums process built on i.i.d. zero mean random 2 2 variables (Xk )k≥0. Let σ = EX1 > 0. Let 0 < α < 1/2 and

p = p(α) := 1/(1/2 − α).

Then −1/2 −1 D 0 n σ ζn −−−→ W in the space H [0, 1] n→∞ α if and only if p (0) lim t P(|X1| ≥ t) = 0. (Lp -condition) t→∞

on the condition.

Necessary and suﬃcient conditions

What is real price that has to be paid to have Lamperti’s invariance o principle in Hα?

(Rouen 2015) Some functional limit theorems 64 / 94 on the condition.

Necessary and suﬃcient conditions

What is real price that has to be paid to have Lamperti’s invariance o principle in Hα? Theorem [NSC for Lamperti IP]

Let ζn be the partial sums process built on i.i.d. zero mean random 2 2 variables (Xk )k≥0. Let σ = EX1 > 0. Let 0 < α < 1/2 and

p = p(α) := 1/(1/2 − α).

Then −1/2 −1 D 0 n σ ζn −−−→ W in the space H [0, 1] n→∞ α if and only if p (0) lim t P(|X1| ≥ t) = 0. (Lp -condition) t→∞

(Rouen 2015) Some functional limit theorems 64 / 94 Necessary and suﬃcient conditions

What is real price that has to be paid to have Lamperti’s invariance o principle in Hα? Theorem [NSC for Lamperti IP]

Let ζn be the partial sums process built on i.i.d. zero mean random 2 2 variables (Xk )k≥0. Let σ = EX1 > 0. Let 0 < α < 1/2 and

p = p(α) := 1/(1/2 − α).

Then −1/2 −1 D 0 n σ ζn −−−→ W in the space H [0, 1] n→∞ α if and only if p (0) lim t P(|X1| ≥ t) = 0. (Lp -condition) t→∞

on the condition.

(Rouen 2015) Some functional limit theorems 64 / 94 Since the supremum giving the H¨oldernorm of a polygonal line is reached at two vertices we have

−1/2 −1/2+α |Sk − Sj | ωα(n ξn, δ) = n max . |k−j|

For n such that 1/n < δ we deduce now

−1/2+α −1/2+α |Sk − Sj | P(n max |Xk | > t) ≤ P(n max > t) ≤ 1≤k≤n 0<|k−j|≤1 |k − j|α

−1/2+α |Sk − Sj | −1/2 P(n max > t) = P(ωα(n ζn, δ) > t) 0<|k−j|

Proof of necessity.

By the continuous mapping theorem, the convergence in distribution −1/2 of n ζn yields for each δ > 0 −1/2 D ωα(n ζn, δ) −−−→ ωα(W , δ). n→∞

(Rouen 2015) Some functional limit theorems 65 / 94 For n such that 1/n < δ we deduce now

−1/2+α −1/2+α |Sk − Sj | P(n max |Xk | > t) ≤ P(n max > t) ≤ 1≤k≤n 0<|k−j|≤1 |k − j|α

−1/2+α |Sk − Sj | −1/2 P(n max > t) = P(ωα(n ζn, δ) > t) 0<|k−j|

Proof of necessity.

By the continuous mapping theorem, the convergence in distribution −1/2 of n ζn yields for each δ > 0 −1/2 D ωα(n ζn, δ) −−−→ ωα(W , δ). n→∞

Since the supremum giving the H¨oldernorm of a polygonal line is reached at two vertices we have

−1/2 −1/2+α |Sk − Sj | ωα(n ξn, δ) = n max . |k−j|

(Rouen 2015) Some functional limit theorems 65 / 94 Proof of necessity.

By the continuous mapping theorem, the convergence in distribution −1/2 of n ζn yields for each δ > 0 −1/2 D ωα(n ζn, δ) −−−→ ωα(W , δ). n→∞

Since the supremum giving the H¨oldernorm of a polygonal line is reached at two vertices we have

−1/2 −1/2+α |Sk − Sj | ωα(n ξn, δ) = n max . |k−j|

For n such that 1/n < δ we deduce now

−1/2+α −1/2+α |Sk − Sj | P(n max |Xk | > t) ≤ P(n max > t) ≤ 1≤k≤n 0<|k−j|≤1 |k − j|α

−1/2+α |Sk − Sj | −1/2 P(n max > t) = P(ωα(n ζn, δ) > t) 0<|k−j|

(Rouen 2015) Some functional limit theorems 65 / 94 By Portmanteau theorem we have

−1/2 lim sup P(ωα(n ζn, δ) > t) ≤ P(ωα(W , δ) > t). n→∞ This yields for each δ > 0

−1/2+α −1/2 lim sup P(n max |Xk | > t) ≤ P(ωα(n ζn, δ) > t). n 1≤k≤n

−1/2 Since P(ωα(n ζn, δ) > t) → 0 as δ → 0 we easily conclude

1/2−δ P( max |Xk | > n t) → 0 as n → ∞. 1≤k≤n

o This is equivalent to the Lp-condition due to independence of Xi ’th.

(Rouen 2015) Some functional limit theorems 66 / 94 Proof of suﬃciency.

The convergence of the ﬁnite dimensional distributions follows by the (0) Donsker invariance principle, since condition Lp -condition yields 2 2 EX1 < ∞. In what follows we assume that EX1 = 1. Since ζn(0) = 0 we have to check the second condition of Theorem tightness of polygonal line processes only. So we have to estimate the probability

αj 1/2 P = P( sup 2 max |S[nr +] − S[nr]| ≥ εn ). J≤j≤log n r∈Dj

(Rouen 2015) Some functional limit theorems 67 / 94 To this aim let δ > 0 be an arbitrary positive number and deﬁne truncated random variables

1/p 0 Xek = Xk χ{|Xk | ≤ n δ}, Xk = Xk − Xek , k = 1,..., n.

Then 1/p 0 P ≤ nP(|X1| ≥ δn ) + P , where

[nr +] 0 αj X 1/2 P = P sup 2 max Xei ≥ (1/3)εn . J≤j≤log n 1≤k≤2j i=[nr]+1

(Rouen 2015) Some functional limit theorems 68 / 94 Since Z ∞ 1/p EXei = EXi χ{|Xi | ≥ δn } = P(|X1| ≥ t)dt ≤ δn1/p Z ∞ p −p −p+1 −1+1/p sup t P(|X1| ≥ t) t dt ≤ cδ n , t δn1/p we have

[nr +] αj −1/2 X sup 2 n max EXei ≤ J≤j≤log n 1≤k≤2j i=[nr]+1 cδ−p+1 sup 2αj 2−j n1/2n−1+1/p ≤ cδ−p+12−J < ε/6 J≤j≤log n for large J.

(Rouen 2015) Some functional limit theorems 69 / 94 Therefore, for n and J large,

[nr +] 0 αj X 1/2 P ≤ P sup 2 max (Xei − EXei ) ≥ (1/6)εn ≤ J≤j≤log n 1≤k≤2j i=[nr]+1 log n u k+1 q −q X qαj −q/2 j X ≤ (ε/6) 2 n 2 E (Xei − EXei ) , j=J i=uk +1 2 where q > 2. By Rosenthal’s inequality (accounting EX1 = 1) u k+1 q X E (Xei − EXei ) ≤ i=uk +1 q/2 2 q/2 q c[(uk+1 − uk ) (EX1 ) + (uk+1 − uk )E|Xe1| ] ≤ c[(n2−j )q/2 + n2−j δq−pn(q−p)/p], since Z δn1/p q q−1 (q−p)/p q−p p E|Xe1| ≤ u P(|X1| > u)du ≤ n δ sup u P(|X1| > u). 0 u

(Rouen 2015) Some functional limit theorems 70 / 94 −q Substituting this estimate we obtain (denoting c0 = c(ε/6) )

log n X P0 ≤ c(ε/6)−q 2qαj n−q/22j [(n2−j )q/2 + n2−j δq−pn(q−p)/p] j=J and choosing q > 1/(1/2 − α) yields

0 −(q(1/2−α)−1)J q−p P ≤ c02 + c0cδ < δ for big J and small δ > 0. The result is proved.

(Rouen 2015) Some functional limit theorems 71 / 94 −q Substituting this estimate we obtain (denoting c0 = c(ε/6) )

log n X P0 ≤ c(ε/6)−q 2qαj n−q/22j [(n2−j )q/2 + n2−j δq−pn(q−p)/p] j=J and choosing q > 1/(1/2 − α) yields

0 −(q(1/2−α)−1)J q−p P ≤ c02 + c0cδ < δ for big J and small δ > 0. The result is proved.

(Rouen 2015) Some functional limit theorems 71 / 94 General functional central limit theorem Deﬁnition

We denote by R the class of functions ρ : [0, 1] → R positive on (0, 1], such that ρ(0) = 0 and satisfying i) for some 0 < α ≤ 1/2, and some function L which is normalized slowly varying at inﬁnity,

ρ(h) = hαL(1/h), 0 < h ≤ 1;

ii) θ(t) = t1/2ρ(1/t) is C 1 on [1, ∞); iii) there is a β > 1/2 and some a > 0, such that θ(t) ln−β(t) is non decreasing on [a, ∞).

Let us recall that L is normalized slowly varying at inﬁnity if and only if for every δ > 0, tδL(t) is ultimately increasing and t−δL(t) is ultimately decreasing. The main practical examples to have in mind is α β ρ(h) = ρα,β(h) := h log (c/h).

(Rouen 2015) Some functional limit theorems 72 / 94 Remark The requirement “for every A > 0” in (NSC) cannot be avoided in general. For instance if X1 is symmetric such that Pr{|X1| ≥ u} = exp(−u/c), (c > 0) and ρ(h) = h1/2 ln(1/h), so that θ(t) = ln t. Clearly (NSC) is satisﬁed only for A > c. Hence, FCLT fails.

o Theorem (FCLT in Hρ)

Let ρ ∈ R. Let ζn be the partial sums process built from i.i.d. zero mean 2 2 random variables (Xk )k≥0. Let σ = EX1 > 0. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if for every A > 0,

1/2 lim tPr |X1| ≥ At ρ(1/t) = 0. (NSC) t→∞

(Rouen 2015) Some functional limit theorems 73 / 94 o Theorem (FCLT in Hρ)

Let ρ ∈ R. Let ζn be the partial sums process built from i.i.d. zero mean 2 2 random variables (Xk )k≥0. Let σ = EX1 > 0. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if for every A > 0,

1/2 lim tPr |X1| ≥ At ρ(1/t) = 0. (NSC) t→∞

Remark The requirement “for every A > 0” in (NSC) cannot be avoided in general. For instance if X1 is symmetric such that Pr{|X1| ≥ u} = exp(−u/c), (c > 0) and ρ(h) = h1/2 ln(1/h), so that θ(t) = ln t. Clearly (NSC) is satisﬁed only for A > c. Hence, FCLT fails.

(Rouen 2015) Some functional limit theorems 73 / 94 Corollary 2

Let ρ(h) = h1/2 lnβ(c/h) with β > 1/2. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if

1/β E exp d|X1| < ∞, for each d > 0.

Corollary 1

Let ρ ∈ R be such that ρ(h) = hαL(1/h) with α < 1/2. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if 1/2 lim tPr |X1| ≥ t ρ(1/t) = 0. t→∞

(Rouen 2015) Some functional limit theorems 74 / 94 Corollary 1

Let ρ ∈ R be such that ρ(h) = hαL(1/h) with α < 1/2. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if 1/2 lim tPr |X1| ≥ t ρ(1/t) = 0. t→∞

Corollary 2

Let ρ(h) = h1/2 lnβ(c/h) with β > 1/2. Then

−1/2 −1 D o n σ ζn −−−→ W in the space H n→∞ ρ if and only if

1/β E exp d|X1| < ∞, for each d > 0.

(Rouen 2015) Some functional limit theorems 74 / 94 What is an idea behind the construction?

Figure: Polygonal line process Figure: Polygonal line process

Adaptive FCLT Various partial sums processes can be built from the sums Sn = X1 + ··· + Xn, n ≥ 0 of independent identically distributed mean zero random variables. Next we focus attention on what we call the adaptive partial sums ad process, denoted ζn .

(Rouen 2015) Some functional limit theorems 75 / 94 Figure: Polygonal line process Figure: Polygonal line process

What is an idea behind the construction?

(Rouen 2015) Some functional limit theorems 75 / 94 Adaptive FCLT Various partial sums processes can be built from the sums Sn = X1 + ··· + Xn, n ≥ 0 of independent identically distributed mean zero random variables. Next we focus attention on what we call the adaptive partial sums ad process, denoted ζn .

What is an idea behind the construction?

Figure:(Rouen 2015)Polygonal line processSome functional limit theoremsFigure: Polygonal line process 75 / 94 ζn(t) 6 Sk+1 ¡ ¡ ¡ ¡ ¡ ¡ Xk ¡ ¡ Sk ¡ - 0 1 k k+1 t n ··· ··· n ··· ··· n ··· 1

Figure: Polygonal line process ζn

(Rouen 2015) Some functional limit theorems 76 / 94 ζn(t) 6 Sk+1 ¡ ¡ ¡ ¡ ¡ ¡ Xk ¡ ¡ Sk ¡ - 0 1 k k+1 t n ··· ··· n ··· ··· n ··· 1

Figure: Correction of polygonal line process ζn

(Rouen 2015) Some functional limit theorems 77 / 94 ζn(t) 6

Sk+1

X k Sk - 0 1 k k+1 t n ··· ··· n ··· ··· n ··· 1

Figure: Polygonal line process ζn

(Rouen 2015) Some functional limit theorems 78 / 94 ζn(t) 6

Sk+1

X k Sk - 0 1 k k+1 t n ··· ··· n ··· ··· n ··· 1

Figure: Corrected polygonal line process ζn

(Rouen 2015) Some functional limit theorems 79 / 94 Let us note, that

2 2 2 Vk Vk−1 Xk 1 E 2 − 2 = E 2 = . Vn Vn Vn n

use stationarity

Adaptive means that the vertices of the corresponding random 2 2 polygonal line have their abscissas at the random points Vk /Vn (0 ≤ k ≤ n) instead of the deterministic equispaced points k/n. By this construction the slope of each line adapts itself to the value of the corresponding random variable. ad By ζn (respectively ζn) we denote the random polygonal partial sums process deﬁned on [0, 1] by linear interpolation between the vertices 2 2 (Vk /Vn , Sk ), k = 0, 1,..., n (respectively (k/n, Sk ), k = 0, 1,..., n), where 2 2 2 Sk = X1 + ··· + Xk , Vk = X1 + ··· + Xk .

For the special case k = 0, we put S0 = 0, V0 = 0.

(Rouen 2015) Some functional limit theorems 80 / 94 Adaptive means that the vertices of the corresponding random 2 2 polygonal line have their abscissas at the random points Vk /Vn (0 ≤ k ≤ n) instead of the deterministic equispaced points k/n. By this construction the slope of each line adapts itself to the value of the corresponding random variable. ad By ζn (respectively ζn) we denote the random polygonal partial sums process deﬁned on [0, 1] by linear interpolation between the vertices 2 2 (Vk /Vn , Sk ), k = 0, 1,..., n (respectively (k/n, Sk ), k = 0, 1,..., n), where 2 2 2 Sk = X1 + ··· + Xk , Vk = X1 + ··· + Xk .

For the special case k = 0, we put S0 = 0, V0 = 0. Let us note, that

2 2 2 Vk Vk−1 Xk 1 E 2 − 2 = E 2 = . Vn Vn Vn n

use stationarity

(Rouen 2015) Some functional limit theorems 80 / 94 Theorem

−1 j 2 −j Assume that ρ ∈ R and limj→∞ j 2 ρ (2 ) = ∞. If X1 is symmetric then V −1ζad −−−→D W , (addaptive IP) n n n→∞ o in Hρ [0, 1] if and only if X1 ∈ DAN.

Corollary

If X1 is symmetric and X1 ∈ DAN then

V −1ζad −−−→D W , n n n→∞

o holds in the space H1/2,β, for any β > 1/2.

It is well known that the Wiener process has a version in the space o H1/2,1/2 but none in H1/2,1/2. Hence Corollary gives the best result possible in the scale of the separable H¨olderspaces Hα,β.

(Rouen 2015) Some functional limit theorems 81 / 94 Theorem Let β > 1/2 and suppose that we have

2 Xk P max 2 ≥ δn −−−→ 0 1≤k≤n Vn n→∞ and V 2 k P max k − ≥ δ −−−→ 0, 2 n 1≤k≤n Vn n n→∞ with

2−(log n)γ 1 δ = c , for some < γ < 1 and some c > 0. n log n 2β Then V −1ζad −−−→D W , in Ho . n n n→∞ 1/2,β

(Rouen 2015) Some functional limit theorems 82 / 94 Conjecture For any β > 1/2 the convergence

V −1ζad −−−→D W n n n→∞

o in the space H1/2,β holds if and only if X1 ∈ DAN and EX1 = 0.

− Observe that n = o(δn) for any > 0. This mild convergence rate δn 2+ε may be obtained as soon as E |X1| is ﬁnite. Corollary

2+ε If for some ε > 0, E |X1| < ∞, then for any β > 1/2

V −1ζad −−−→D W in the space Ho . n n n→∞ 1/2,β

This result contrasts strongly with the extension of Lamperti’s invariance principle in the same functional framework.

(Rouen 2015) Some functional limit theorems 83 / 94 − Observe that n = o(δn) for any > 0. This mild convergence rate δn 2+ε may be obtained as soon as E |X1| is ﬁnite. Corollary

2+ε If for some ε > 0, E |X1| < ∞, then for any β > 1/2

V −1ζad −−−→D W in the space Ho . n n n→∞ 1/2,β

This result contrasts strongly with the extension of Lamperti’s invariance principle in the same functional framework. Conjecture For any β > 1/2 the convergence

V −1ζad −−−→D W n n n→∞

o in the space H1/2,β holds if and only if X1 ∈ DAN and EX1 = 0.

(Rouen 2015) Some functional limit theorems 83 / 94 FCLT for triangular arrays In this lecture we tackle the case of triangular arrays

(Xn,k , k = 1,..., kn), n = 1, 2,...,

of row wise independent random variables. We assume in all the sequel 2 2 that EXn,k = 0, σn,k := var(Xn,k ) = EXn,k < ∞ and σn,k > 0 for k = 1,..., kn, n = 1, 2,... Put Sn(0) = 0, bn(0) = 0 and

2 2 Sn(k) = Xn,1 + ··· + Xn,k , bn(k) = σn,1 + ··· + σn,k ,

for k = 1, 2,..., kn. We also assume that bn(kn) = 1. Consider the process Ξn(t), t ∈ [0, 1], built by aﬃne interpolation between the points (bn(k), Sn(k)), k = 0, 1,..., kn:

−2 Ξn(t) = Sn(k − 1) + σn,k (t − bn(k))Xn,k , bn(k − 1) ≤ t ≤ bn(k),

k = 1,..., kn.

(Rouen 2015) Some functional limit theorems 84 / 94 In the ﬁrst H¨olderianfunctional central limit theorem we assume the existence of moments of order q > 2 and generalize Lamperti’s theorem to the case of triangular arrays. Theorem [IP with moments] If 2 lim max σn,k = 0 n→∞ 1≤k≤kn and if for some q > 2,

kn X −2αq q lim σ E |Xn,k | = 0, n→∞ n,k k=1 then D o Ξn −−−→ W in the space H , n→∞ α for any α such that 0 ≤ α < 1/2 − 1/q.

(Rouen 2015) Some functional limit theorems 85 / 94 In order to relax the moment condition let us introduce for each τ > 0, the truncated random variables

2α Xn,k,τ := Xn,k 1{|Xn,k | ≤ τσn,k }, k = 1,..., kn; n = 1, 2,...

Theorem [IP for triangular array] Assume that the following conditions are satisﬁed.

1 Pkn 2α For every ε > 0, limn→∞ k=1 Pr(|Xn,k | ≥ εσn,k ) = 0.

2 Pkn 2 For every ε > 0, limn→∞ k=1 EXn,k 1{|Xn,k | ≥ ε} = 0. −2αq 3 Pkn q For some q > p(α), limτ→0 lim supn→∞ k=1 σn,k E |Xn,k,τ | = 0. Then D o Ξn −−−→ W in the space H , n→∞ α for any α such that 0 ≤ α < 1/2 − 1/q.

(Rouen 2015) Some functional limit theorems 86 / 94 Let us consider now the more speciﬁc case of a triangular array whose rows are linear combinations with deterministic coeﬃcients of the same i.i.d. sequence.

Let (Xk ) be a sequence of i.i.d. centered random variables, with 2 variance EXk = 1. Let a := (an,k , 1 ≤ k ≤ kn, n ≥ 1) be a triangular array of non null real numbers satisfying

kn X 2 ∀n ≥ 1, an,k = 1. k=1 and 2 lim max an,k = 0. n→∞ 1≤k≤kn

(Rouen 2015) Some functional limit theorems 87 / 94 Let us denote by Ξa,n the polygonal partial sums process built on the triangular array (Xn,k ), where

Xn,k := an,k Xk , k = 1,..., kn, n ≥ 1.

Theorem (IP for weighted partial sums) Assume that the following conditions are satisﬁed.

kn X 2α−1 for all ε > 0, lim Pr |X1| ≥ ε|an,k | = 0. n→∞ k=1

kn p(α) X 2α−1 M := sup t Pr |X1| ≥ t|an,k | < ∞. t>0, n≥1 k=1 Then D o Ξa,n −−−→ W in the space H . n→∞ α

(Rouen 2015) Some functional limit theorems 88 / 94 Corollary

Pkn 2 Assume that the triangular array (an,k ) satisﬁes k=1 ank = 1 and the sequence (kn)n≥1 is non decreasing, such that k sup n+1 < ∞ n≥1 kn and β 2 γ ∀n ≥ 1, ∀k = 1,..., kn, ≤ an,k ≤ , kn kn for some positive constants β, γ. Then Ξa,n converges in distribution to o W in the space Hα if and only if

p(α) lim t Pr(|X1| ≥ t) = 0. t→∞

(Rouen 2015) Some functional limit theorems 89 / 94 Bootstrap invariance principle

Let X1,..., Xn be random variables deﬁned on the same probability space (Ω, F, Pr), independent, with null expectation and having the same distribution function F .

We write Fbn for the corresponding empirical distribution function: −1 Pn Fbn(x) = n k=1 1{Xk ≤ x}, x ∈ R. ∗ ∗ Let X1 ,..., Xn be a bootstraped sample of Fbn, i.e. conditionally to ∗ ∗ X1,..., Xn, the X1 ,..., Xn are independent random variables with the same distribution function Fbn. ∗ We denote by E the conditional expectation given X1,..., Xn. ∗ ∗ −1 P ∗ Put Sn (0) := 0, Sn (t) := σbn j≤t (Xj − X n), 0 < t ≤ n, with the classical notations

n −1 2 −1 X 2 X n := n (X1 + ··· + Xn) and σbn := n (Xk − X n) . k=1

(Rouen 2015) Some functional limit theorems 90 / 94 ∗ Consider the polygonal line process ζn (t), t ∈ [0, 1], with vertices at ∗ the points k/n, Sn (k) , k = 0, 1,..., n. This polygonal line being doubly random, it is suitable to precise the convergence mode under consideration when studying its asymptotic behaviour. Denote by d some metric inducing the convergence in distribution in o Hα.

(Rouen 2015) Some functional limit theorems 91 / 94 Theorem [Bootstrap IP a.s.]

The moment condition p(α) E |X1| < ∞ is necessary and suﬃcient for the convergence

d(n−1/2ζ∗, W ) −−−→a.s. 0. n n→∞

Theorem [Bootstrap IP in probability]

The condition p(α) lim t Pr(|X1| > t) = 0 t→∞ is necessary and suﬃcient for the convergence

d(n−1/2ζ∗, W ) −−−→Pr 0. n n→∞

(Rouen 2015) Some functional limit theorems 92 / 94 Proofs

To establish the suﬃciency, we use Theorem IP for triangular array, conditionally to X1,..., Xn, putting

−1/2 −1 ∗ Xn,k := n σbn (Xk − X n), k = 1,..., n. 2 ∗ ∗2 −1 Here σn,k = E Xn,k = n , which reduces the proof to that of the convergence qα−q/2 −q ∗ ∗ q a.s. n σ nE |X − X n| −−−→ 0, bn 1 n→∞ 2 for some q > p(α) > 2. As EX1 is ﬁnite, the strong law of large numbers gives a.s. 2 a.s. X n −−−→ 0 and σ −−−→ 1. n→∞ bn n→∞

(Rouen 2015) Some functional limit theorems 93 / 94 Next we have to show

n qα−q/2 ∗ ∗ q −q/p(α) X q a.s. n nE |X | = n |Xk | −−−→ 0. 1 n→∞ k=1 According to Marcinkiewicz-Zygmund strong law of large numbers, if β (Yk )k≥1 is an i.i.d. sequence such that E |Y1| < ∞ for some β ∈]0, 1[, −1/β then n (Y1 + ··· + Yn) converges almost surely to 0. Choosing q Yk = |Xk | and β = p(α)/q ends the proof.

(Rouen 2015) Some functional limit theorems 94 / 94 Next we have to show

(Rouen 2015) Some functional limit theorems 94 / 94