Necessary and sufficient condition for Lamperti’s invariance principle^{∗}

Alfredas Račkauskas

Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius Lithuania

Charles Suquet

Statistique et Probabilités E.P. CNRS 1765, Bât.M2 Université des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex France

1 Introduction and results

Let X_{1}, …, X_{n}, … are i.i.d. mean zero random variables. Set S_{0} = 0,

S_{k} = X_{1} + ⋯ + X_{k}, for k = 1, 2, …

and consider the process

ξ_{n}(t) = S_{[nt]} + (nt − [nt])X_{[nt]+1}, t ∈ [0, 1].

Classical Donsker-Prohorov invariance principle states, that if EX_{1}^{2} = 1, then

n^{−1/2}ξ_{n} ⟶^{𝒟} W,  (1)

in C[0, 1], where (W (t), t ∈ R) is a standard Wiener process and ⟶^{𝒟} denotes convergence in distribution. Since (1) yields the central limit theorem, the finiteness of the second moment of X_{1} therefore is necessary.

Lamperti [5] considered the convergence (1) with respect to a stronger Hölderian topology. Let us recal his result. For 0 < α ≤ 1, let H_{α}[0, 1] be the set of real valued continuous functions x : [0, 1] → ℝ such that ω_{α}(x, 1) < ∞, where

∗Research supported by a cooperation agreement CNRS/LITHUANIA (4714)

ω_{α}(x, δ) = sup_{t,s∈[0,1], 0<|t−s|<δ} \frac{|x(t) − x(s)|}{|s − t|^{α}}.

The set H_{α}[0, 1] is a Banach space when endowed with the norm ||x||_{α} = |x(0)| + ω_{α}(x, 1) and

H_{α}^{0}[0, 1] = {x ∈ H_{α}([0, 1]) : lim_{δ→0} ω_{α}(x, δ) = 0}.

is its closed separable subspace. Lamperti [5] proved that if 0 < α < 1/2 and E|X_{1}|^{p} < ∞, where p > 1/(1/2−α), then (1) takes place in the Hölder space H_{α}^{0}[0, 1]. This result was derived again by Kerkyacharian and Roynette [4] by another method using Ciesielski [1] analysis of Hölder spaces by triangular functions. Further generalizations were given by Ericson [2], Hamadouche [3], Račkauskas and Suquet [6]. Lamperti [5] also noticed, that if (1) holds in H_{α}[0, 1], 0 < α < 1/2 then necessarily E|X_{1}|^{p} < ∞ for any p < 1/(1/2 − α).

The main result of this notes is the following

Theorem 1.1 Assume that 0 < α < 1/2 and p = 1/(1/2 − α). Then

n^{−1/2}ξ_{n} ⟶_{n→∞}^{𝒟} W in the space H_{α}^{0}[0, 1]

if and only if

lim_{t→∞} t^{p}P (|X_{1}| ≥ t) = 0.  (2)

Corollary 1.1 Let 0 < α < 1/2 and p = 1/(1/2 − α). Then for each δ > 0

n^{−1/2+α} max_{0<|k−j|<δn} \frac{|S_{k} − S_{j}|}{|k − j|^{α}} ⟶_{n→∞}^{𝒟} sup_{0<|t−s|<δ} \frac{|W (s) − W (t)|}{|s − t|^{α}}

if and only if the condition (2) holds.

2 Preliminaries

Let

Λ(t) = \begin{cases}2t & if 0 ≤ t ≤ 1/2 \\ 2(1 − t) & if 1/2 < t ≤ 1 \\ 0 & otherwise. \end{cases}

For n = 2^{j} + k, 0 ≤ k < 2^{j}, j = 0, 1, … set

Λ_{n}(t) = Λ_{jk}(t) = Λ(2^{j}t − k), t ∈ [0, 1],

Let Λ_{−1}(t) = χ_{[0,1]}(t) and Λ_{0}(t) = Λ_{−1,0}(t) = t. Then the family {Λ_{n}, n ≥ −1} is a Schauder basis on the space H_{α}^{0}[0, 1]: each function x ∈ H_{α}^{0}[0, 1] admits the series representation

x(t) = x(0) + ∑_{n=0}^{∞} λ_{n}(x)Λ_{n}(t), t ∈ [0, 1],

where λ_{−1}(x) = λ_{−2,0}(x) = x(0), λ_{0}(x) = λ_{−1,0}(x) = x(1) − x(0), and for n = 2^{j} + k, j ≥ 0, 0 ≤ k < 2^{j} :

λ_{n}(x) = λ_{jk}(x) = x((k + 1/2)2^{−j}) − \frac{1}{2}\{x(k2^{−j}) + x((k + 1)2^{−j})\}.

By Ciesielski (1960), the norm ||x||_{α} is equivalent to the sequence norm

||x||_{α}^{seq} = sup_{j≥−2} 2^{(j+1)α} max_{0≤k<2^{j}} |λ_{jk}(x)|.

The tightness criterion we shall exploit for the proof of the main result is the following one.

Theorem 2.1 The sequence (ζ_{N}) of random elements in the Hölder space H_{α}^{0}[0, 1] is tight if and only if the following two conditions are satisfied:

i) lim_{b→∞} sup_{N} P {|ζ_{N}(0)| > b} = 0;

ii) for each ε > 0

lim_{J→∞} lim sup_{N} P {sup_{i≥J} 2^{αj} max_{0≤k<2^{j}} λ_{jk}(ζ_{N}) > ε} = 0.

For the proof see ...

3 Proofs

3.1 Proof of theorem 1.1

Necessity. By continuous mapping theorem, the convergence in distribution of n^{−1/2}ξ_{n} yields for each δ > 0

ω_{α}(n^{−1/2}ξ_{n}, δ) ⟶^{𝒟} ω_{α}(W, δ).  (3)

Since

ω_{α}(n^{−1/2}ξ_{n}, δ) = n^{−1/2+α} max_{|k−j|<nδ} \frac{|S_{k} − S_{j}|}{|k − j|^{α}}

we have for n such that 1/n < δ

P (n^{−1/2+α} max_{1≤k≤n} |X_{k}| > t) ≤ P (n^{−1/2+α} max_{0<|k−j|≤1} \frac{|S_{k} − S_{j}|}{|k − j|^{α}} > t) ≤

P (n^{−1/2+α} max_{0<|k−j|<nδ} \frac{|S_{k} − S_{j}|}{|k − j|^{α}} > t) = P (ω_{α}(n^{−1/2}ξ_{n}, δ) > t)

By the convergence (3)

lim sup_{n→∞} P (ω_{α}(n^{−1/2}ξ_{n}, δ) > t) ≤ P (ω_{α}(W, δ) > t).

This yields for each δ > 0

lim sup_{n} P (n^{−1/2+α} max_{1≤k≤n} |X_{k}| > t) ≤ P (ω_{α}(n^{−1/2}ξ_{n}, δ) > t).

Since P (ω_{α}(n^{−1/2}ξ_{n}, δ) > t) → 0 as δ → 0 we easily conclude the necessity of the condition (2).

Sufficiency. The convergence of the finite dimensional distributions follows by the Donsker-Prohorov invariance principle, since the condition (2) yields EX_{1}^{2} < ∞. In what follows we assume that EX_{1}^{2} = 1.

To prove tightness we shall use theorem 2.1. Throughout t_{k} = t_{kj} = k2^{−j}, k = 0, 1, …, 2^{j}, j = 1, 2, … Since ξ_{n}(0) = 0 we have to check the second condition of the theorem 2.1 only. To this aim we have to estimate

P = P (sup_{J≤j} 2^{αj}n^{−1/2} max_{1≤k<2^{j}} |ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| ≥ ε) ≤ P_{1} + P_{2},

where

P_{1} = P ( sup_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} |ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| ≥ ε)

and

P_{2} = P ( sup_{j>log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} |ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| ≥ ε).

Estimation of P_{2}. If j > log n then t_{k+1}−t_{k} = (k+1)2^{−j}−k2^{−j} = 2^{−j} < 1/n and therefore if t_{k} ∈ [v/n, (v + 1)/n) then either t_{k+1} ∈ [v/n, (v + 1)/n) or t_{k+1} ∈ [(v + 1)/n, (v + 2)/n), where 1 ≤ v ≤ n − 2 depends on k and j. If both t_{k} and t_{k+1} are in the same interval, then we have

|ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| = |X_{v+1}|2^{−j}n ≤ 2^{−j}n max_{1≤v≤n} |X_{v}|.

If t_{k} and t_{k+1} are in the different intervals, then

|ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| ≤ |ξ_{n}(t_{k}) − ξ_{n}((v + 1)/n)| +

|ξ_{n}((v + 1)/n) − ξ_{n}(t_{k+1})| ≤ 22^{−j}n max_{1≤v≤n} |X_{v}|.

With both cases taken into account we obtain

P_{2} ≤ P ( max_{j>log n} 2^{αj}n2^{−j}n^{−1/2} max_{1≤v≤n} |X_{v}| ≥ ε/2) ≤

P (n^{α−1/2} max_{1≤v≤n} |X_{v}| ≥ ε/2) ≤ nP (|X_{1}| ≥ (ε/2)n^{1/p}).

Hence, due to the condition (2), for each ε > 0 lim_{n} P_{2} = 0. Estimation of P_{1}. Let u_{k} = [nt_{k}]. Then u_{k} ≤ nt_{k} ≤ u_{k} + 1 and u_{k} + 1 ≤ u_{k+1} ≤ nt_{k+1} ≤ u_{k+1} + 1. Therefore

|ξ_{n}(t_{k+1}) − ξ_{n}(t_{k})| ≤
|ξ_{n}(t_{k+1}) − S_{u_{k+1}}| + |S_{u_{k+1}} − S_{u_{k}}| + |S_{u_{k}} − ξ_{n}(t_{k})|.

Since |S_{u_{k}} − ξ_{n}(t_{k})| ≤ |X_{u_{k}+1}| and |ξ_{n}(t_{k+1}) − S_{u_{k+1}}| ≤ |X_{u_{k+1}+1}| we obtain P_{1} ≤ P_{11} + P_{12} where

P_{11} = P ( sup_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} |S_{u_{k+1}} − S_{u_{k}}| ≥ ε/3)

P_{12} = P ( max_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤v≤n} |X_{v}| ≥ ε/6) ≤

P (n^{α−1/2} max_{1≤v≤n} |X_{v}| ≥ ε/6) ≤ nP (|X_{1}| > n^{1/p}ε/6).

Since P_{12} → 0 as n → ∞ due to the condition (2), we have to deal with P_{11} only. To this aim let δ > 0 be an arbitrary number and define the truncated random variables

X̃_{k} = X_{k}χ{|X_{k}| ≤ n^{1/p}δ}, X′_{k} = X_{k} − X̃_{k}, k = 1, …, n.

Then

P_{11} ≤ nP (|X_{1}| ≥ δn^{1/p}) + P′_{11},

where

P′_{11} = P ( sup_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} | ∑_{i=u_{k}+1}^{u_{k+1}} X̃_{i}| ≥ ε/3).

Since EX̃_{i} = EX_{i}χ{|X_{i}| ≥ δn^{1/p}} = ∫_{δn^{1/p}}^{∞} P (|X_{1}| ≥ t)dt ≤

sup_{t} t^{p}P (|X_{1}| ≥ t) ∫_{δn^{1/p}}^{∞} t^{−p}dt ≤ cδ^{−p+1}n^{−1+1/p},

we have

sup_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} | ∑_{i=u_{k}+1}^{u_{k+1}} EX̃_{i}| ≤

cδ^{−p+1} sup_{J≤j≤log n} 2^{αj}2^{−j}n^{1/2}n^{−1+1/p} ≤ cδ^{−p+1}2^{−J} < ε/6

for large J. Therefore for n and J large

P′_{11} ≤ P ( sup_{J≤j≤log n} 2^{αj}n^{−1/2} max_{1≤k≤2^{j}} | ∑_{i=u_{k}+1}^{u_{k+1}}(X̃_{i} − EX̃_{i})| ≥ ε/6) ≤

≤ (ε/6)^{−q} ∑_{j=J}^{log n} 2^{qαj}n^{−q/2}2^{j}E| ∑_{i=u_{k}+1}^{u_{k+1}}(X̃_{i} − EX̃_{i})|^{q},

where q > 2. By Rosenthal’s inequality (accounting EX_{1}^{2} = 1)

E| ∑_{i=u_{k}+1}^{u_{k+1}}(X̃_{i} − EX̃_{i})|^{q} ≤

c[(u_{k+1} − u_{k})^{q/2}(EX_{1}^{2})^{q/2} + (u_{k+1} − u_{k})E|X̃_{1}|^{q}] ≤ c[(n2^{−j})^{q/2} + n2^{−j}δ^{q−p}n^{(q−p)/p}],

since

E|X̃_{1}|^{q} ≤ ∫_{0}^{δn^{1/p}} u^{q−1}P (|X_{1}| > u)du ≤ n^{(q−p)/p}δ^{q−p} sup_{u} u^{p}P (|X_{1}| > u).

Substituting this estimate we obtain (denoting c_{0} = c(ε/6)^{−q})

P′_{11} ≤ c(ε/6)^{−q} ∑_{j=J}^{log n} 2^{qαj}n^{−q/2}2^{j}[(n2^{−j})^{q/2} + n2^{−j}δ^{q−p}n^{(q−p)/p}] ≤

c(ε/6)^{−q} ∑_{j=J}^{log n} 2^{qαj}n^{−q/2}2^{j}(n2^{−j})^{q/2} +

c(ε/6)^{−q}δ^{q−p} ∑_{j=J}^{log n} 2^{qαj}n^{−q/2}2^{j}n2^{−j}n^{(q−p)/p} ≤

c(ε/6)^{−q} ∑_{j=J}^{log n} 2^{qαj+1−q/2} + c(ε/6)^{−q}δ^{q−p}n^{−qα} ∑_{j=J}^{log n} 2^{qαj}.

Choosing q > 1/(1/2 − α) we obtain

P′_{11} ≤ c_{0}2^{−(q(1/2−α)−1)J} + c_{0}cδ^{q−p} < δ

for big J. The result is proved.

3.2 Proof of corollary 1.1

Sufficiency follows by theorem 1.1 and continuous mapping theorem. Necessity follows from the proof of theorem 1.1.

References

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[3] Hamadouche, D. (1998). Invariance principles in Hölder spaces (preprint). Pub. IRMA Lille 44-VII.

[4] Kerkyacharian, G. and Roynette, B. (1991). Une démonstration simple des théorèmes de Kolmogorov, Donsker et Ito-Nisio. C. R. Acad. Sci. Paris Série I 312, 877–882.

[5] Lamperti, J. (1962). On convergence of stochastic processes. Trans. Amer. Math. Soc. 104, 430–435.

[6] Račkauskas, A. and Suquet, Ch. (1998). On the H”olderian functional central limit theorem for i.i.d. random elements in Banach space(preprint). Pub. IRMA Lille 50-III.