PUB. IRMA, LILLE 2001 Vol. 53, N^{o} VI

Convergence of self-normalized partial sums processes in C[0, 1] and D[0, 1]^{∗}

Alfredas Račkauskas

Vilnius university and Institute of Mathematics and Informatics

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

Charles Suquet Université des Sciences et Technologies de Lille

Statistique et Probabilités F.R.E. CNRS 2222 Bât. M2, U.F.R. de Mathématiques F-59655 Villeneuve d’Ascq Cedex France

Abstract

Let (X_{i})_{i≥1} be an i.i.d. sequence of mean zero random variables, S_{n} := X_{1} + ⋯ + X_{n} and V_{n}^{2} := X_{1}^{2} + ⋯ + X_{n}^{2}. We consider four sequences of partial sums processes: the broken lines with vertices at the points (k/n, S_{k}/V_{n}) or (V_{k}^{2}/V_{n}^{2}, S_{k}/V_{n}) and the corresponding random step functions. We prove that each of them converges weakly in C[0, 1] or D[0, 1] to the Brownian motion if and only if X_{1} belongs to the domain of attraction of the normal distribution. These results contrast with the classical Donsker Prohorov invariance principles where the N.S.C. for such convergences is E X_{1}^{2} < ∞.

Résumé

Soit (X_{i})_{i≥1} une suite de variables aléatoires i.i.d. centrées, S_{n} := X_{1} + ⋯ + X_{n} et V_{n}^{2} := X_{1}^{2} + ⋯ + X_{n}^{2}. Nous considérons quatre suites de processus de sommes partielles : les lignes polygonales de sommets les points (k/n, S_{k}/V_{n}) ou (V_{k}^{2}/V_{n}^{2}, S_{k}/V_{n}) et les fonctions aléatoires étagées correspondantes. Nous prouvons que chacune de ces suites converge en loi dans C[0, 1] ou D[0, 1] vers le mouvement brownien si et seulement si X_{1} est dans le domaine d’attraction de la loi normale. Ces résultats contrastent avec les principes d’invariance de Donsker Prohorov classiques pour lesquels la C.N.S. est E X_{1}^{2} < ∞.

Keywords: domain of attraction, functional central limit theorem, invariance principle, self-normalization. AMS classifications: 60F05, 60B05, 60G17, 60E10.

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

VI – 3

1 Introduction

Many limit theorems in the theory of probability deal with the asymptotic behavior of sums S_{n} = X_{1}+⋯+X_{n} of independent identically distributed mean zero random variables, normalized by some deterministic sequence (b_{n}). When the X_{i}’s are square integrable, the classical choice of b_{n} = \sqrt{n} is used to obtain central limit theorems as well as invariance principles in C[0, 1] or D[0, 1]. Selfnormalization consists here in replacing b_{n} by V_{n} = (X_{1}^{2} + ⋯ + X_{n}^{2})^{1/2}. There is of course a strong statistical interest in this replacement. For instance the classical Student t-statistic can be easily expressed in term of self-normalized sums. It is now well established that in general self-normalization improves the asymptotic properties of sums of independent random variables.

A rich litterature is devoted to limit theorems for self-normalized sums. Logan, Mallows, Rice and Shepp [11] investigate the various possible limit distributions of self-normalized sums. Giné, Götze and Mason [8] prove that S_{n}/V_{n} converges to the Gaussian standard distribution if and only if X_{1} is in “DAN”, the domain of attraction of the normal distribution (the symetric case was previously treated in Griffin and Mason [10]). Egorov [7] investigates the non identically distributed case. Bentkus and Götze [3] obtain the rate of convergence of S_{n}/V_{n} when X_{1} ∈ DAN. Griffin and Kuelbs [9] prove the LIL for self-normalized sums when X_{1} ∈ DAN. Moderate deviations (of Linnik’s type) are studied in Shao [15] and Christiakov and Götze [5]. Large deviations (of Cramér-Chernoff type) are investigated in Shao [14] without moment conditions.

Chuprunov [6] gives invariance principles for various partial sums processes under self-normalization in C[0, 1] or D[0, 1]. Račkauskas and Suquet [13] obtained the weak convergence in the best possible Hölder space of the adaptive broken line of self-normalized partial sums of symetric i.i.d. X_{i}’s when X_{1} ∈ DAN. In the present contribution we are concerned with invariance principles in the more classical frameworks of C[0, 1] or D[0, 1] for processes of self-normalized partial sums of i.i.d. X_{i}’s in DAN, without symetry assumption. We prove that these processes converge to the Brownian motion in C[0, 1] or D[0, 1] if and only if X_{1} ∈ DAN. This improves on Chuprunov’s result in the i.i.d. case.

Let us fix now the notations used throughout the paper. X_{1}, …, X_{n}, … are i.i.d. mean zero random variables. Set S_{0} = 0, V_{0} = 0,

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

When V_{n} = 0, we adopt the convention that S_{k}/V_{n} := 0 (k ≤ n). We consider the following processes:

• the standard (non normalized) broken line process

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

• the adaptive broken line process ζ_{n} = (ζ_{n}(t), t ∈ [0, 1]) which is defined by linear interpolation of the points

(V_{k}^{2}/V_{n}^{2}, S_{k}/V_{n}), k = 0, …, n,

whenever V_{n} > 0 and by ζ_{n} = 0 when V_{n} = 0;

• the standard (non normalized) step process

W_{n}(t) = S_{[nt]}, t ∈ [0, 1];

• the adaptive step process Z_{n} = (Z_{n}(t), t ∈ [0, 1]) which whenever V_{n} > 0 is constant on each intervall [V_{k}^{2}/V_{n}^{2}, V_{k+1}^{2}/V_{n}^{2}) and satisfies

Z_{n}(\frac{V_{k}^{2}}{V_{n}^{2}} ) = \frac{S_{k}}{V_{n}} k = 0, …, n,

while for V_{n} = 0, Z_{n} := 0.

2 Donsker-Prohorov invariance principle under self-normalization

Before stating the results let us remind that X_{1} ∈ DAN means that there exists a sequence ℓ_{n} ↑ ∞ such that

ℓ_{n}^{−1}n^{−1/2}S_{n} ⟶^{𝒟} N(0, 1).  (1)

This yields that

ℓ_{n}^{−2}n^{−1}V_{n}^{2} ⟶^{P} 1.  (2)

We have moreover for each τ > 0,

nP (|X_{1}| > τℓ_{n}\sqrt{n}) → 0,  (3)
ℓ_{n}^{−2}E (X_{1}^{2}; |X_{1}| ≤ τℓ_{n}\sqrt{n}) → 1,  (4)
nE (X_{1}; |X_{1}| ≤ τℓ_{n}\sqrt{n}) → 0,  (5)

see for instance Araujo and Giné [1], Chap. 2, Cor. 4.8 (a), Th. 6.17 (i) and Cor. 6.18 (b). Here and in all the paper (X; E) means the product of the random variable X by the indicator function of the event E.

Let us also recall O’Brien’s [12] result: X_{1} ∈ DAN if and only if

V_{n}^{−1} max_{1≤k≤n} |X_{k}| ⟶^{P} 0.  (6)

Throughout the paper, W denotes a standard Brownian motion.

Theorem 2.1 The convergence

\frac{ξ_{n}}{V_{n}} ⟶^{𝒟} W  (7)

holds in the space C[0, 1] if and only if X_{1} ∈ DAN.

Theorem 2.2 The convergence

V_{n}^{−1}ζ_{n} ⟶^{𝒟} W  (8)

holds in the space C[0, 1] if and only if X_{1} ∈ DAN.

Let us remark that the necessity of X_{1} ∈ DAN in both Theorem 2.1 and Theorem 2.2 follows from Giné, Götze and Mason [8]. Let us notice also that (1) or (8) both exclude the trivial case P (X_{1} = 0) = 1, so that almost surely V_{n} > 0 for large enough n.

Proof of theorem 2.1. First we prove the convergence of finite dimensional distributions (f.d.d.) of the process (V_{n}^{−1}ξ_{n}) to the corresponding f.d.d. of the Wiener process W.

To this aim, consider the process W_{n} = (S_{[nt]}, t ∈ [0, 1]). By (6) applied to the obvious bound

sup_{0≤t≤1} V_{n}^{−1}|ξ_{n}(t) − W_{n}(t)| ≤ V_{n}^{−1} max_{1≤k≤n} |X_{k}|,

the convergence of f.d.d. of V_{n}^{−1}ξ_{n} follows from those of the process V_{n}^{−1}W_{n}.

Let 0 ≤ t_{1} < t_{2} < ⋯ < t_{d} ≤ 1 and let (ℓ_{n}) be the (non random) sequence introduced above. From (1), independence of the X′_{i}s and the fact that ℓ_{[nt]}^{2}ℓ_{n}^{−2} converges to 1 (see e.g. Remark 3.2 in [13]), we get

ℓ_{n}^{−1}n^{−1/2}(S_{[nt_{1}]}, S_{[nt_{2}]} − S_{[nt_{1}]}, …, S_{[nt_{d}]} − S_{[nt_{d−1}]}) ⟶^{𝒟}

(W (t_{1}), W (t_{2}) − W (t_{1}), …, W (t_{d}) − W (t_{d−1})).

Now (2) and the continuity of the map

(x_{1}, x_{2}, …, x_{d}) ↦ (x_{1}, x_{2} + x_{1}, …, x_{d} + ⋯ + x_{1})

yields the convergence of f.d.d. of V_{n}^{−1}W_{n}. The convergence of finite dimensional distributions of the process V_{n}^{−1}ξ_{n} is thus established.

To prove the tightness we shall use Theorem 8.3 from Billingsley [4]. Since ξ_{n}(0) = 0, the proof reduces in showing that for all ε, η > 0 there exist n_{0} ≥ 1 and δ, 0 < δ < 1, such that

\frac{1}{δ}P \{ sup_{1≤i≤nδ} V_{n}^{−1}|S_{k+i} − S_{k}| ≥ ε\} ≤ η, n ≥ n_{0},  (9)

for all 1 ≤ k ≤ n.

Let us introduce the truncated variables

Y_{i} := ℓ_{n}^{−1}(X_{i}; X_{i}^{2} ≤ τ^{2}ℓ_{n}^{2}n), i = 1, …, n,

where ℓ_{n} is defined above and τ will be chosen later. Denote by S̃_{k} and Ṽ_{k} the corresponding partial sums with their autonormalizing random variables:

S̃_{k} = Y_{1} + ⋯ + Y_{k}, Ṽ_{k} = (Y_{1}^{2} + ⋯ Y_{n}^{2})^{1/2}, k = 1, …, n.

Then we have

P { sup_{1≤i≤nδ} V_{n}^{−1}|S_{k+i} − S_{k}| ≥ ε} ≤ A + B + C,  (10)

where

A := P \{ sup_{1≤i≤nδ} |S̃_{k+i} − S̃_{k}| ≥ ε\sqrt{n}/2\},
B := P \{Ṽ_{n} < \sqrt{n}/2\},
C := nP \{|X_{1}| ≥ τℓ_{n}\sqrt{n}\}.

Due to (5) we can choose n_{1} such that \sqrt{n}|E Y_{1}| ≤ 1/4 for n ≥ n_{1}. Then with n ≥ n_{1} and δ ≤ ε we have

A ≤ P \{ max_{1≤i≤nδ}| ∑_{j=k+1}^{k+i}(Y_{j} − E Y_{j})| + nδ|E Y_{1}| ≥ \sqrt{n}ε/2\}

≤ P \{ max_{1≤i≤nδ}| ∑_{j=k+1}^{k+i}(Y_{j} − E Y_{j})| ≥ \sqrt{n}ε/4\}.

By Chebyshev’s inequality and Rosenthal inequality with p > 2, we have for each 1 ≤ k ≤ nδ

P \{n^{−1/2}| ∑_{j=k+1}^{k+nδ}(Y_{j} − E Y_{j})| ≥ \frac{ε}{8}\} ≤ \frac{8^{p}}{ε^{p}n^{p/2}} E | ∑_{j=k+1}^{k+nδ}(Y_{j} − E Y_{j})|^{p}

≤ \frac{8^{p}}{ε^{p}n^{p/2}} [(nδ)^{p/2}(E Y_{1}^{2})^{p/2} + nδE |Y_{1}|^{p}].

By (4) we can choose n_{2} such that

3/4 ≤ E Y_{1}^{2} ≤ 3/2 for n ≥ n_{2}.  (11)

Then we have E |Y_{1}|^{p} ≤ 2n^{(p−2)/2}τ^{p−2} and then assuming that τ ≤ δ^{1/2} we obtain

P \{n^{−1/2}| ∑_{j=k+1}^{k+nδ}(Y_{j} − E Y_{j})| ≥ \frac{ε}{8}\} ≤ \frac{8^{p}}{ε^{p}n^{p/2}} [2^{p/2}(nδ)^{p/2} + δn^{p/2}τ^{p−2}]

≤ \frac{2 · 16^{p}δ^{p/2}}{ε^{p}} .

Now by Ottaviani inequality we find

A ≤ \frac{δη}{3} ,  (12)

provided δ^{p/2} ≤ ε^{p}/(4 · 16^{p}) and δ^{(p−2)/2} ≤ ηε^{p}/(6 · 16^{p}).

Next we consider B. Since n^{−1}E Ṽ_{n}^{2} = E Y_{1}^{2} we have by (11) n^{−1}E Ṽ_{n}^{2} ≥ 3/4, for n ≥ n_{2}. Furthermore

B ≤ P {n^{−1}|Ṽ_{n}^{2} − E Ṽ_{n}^{2}| ≥ 1/2} ≤ 4n^{−1}E Y_{1}^{4} ≤ 4τ^{2}E Y_{1}^{2} ≤ δη/3,  (13)

provided n ≥ n_{2} and τ^{2} ≤ δη/18.

Finally choose n_{3} such that C ≤ ηδ/3 when n ≥ n_{3} and join to that the estimates (12, 13) to conclude (9). The proof is completed.

Proof of theorem 2.2.

Due to Theorem 2.1, it suffices to check that ∥V_{n}^{−1}ξ_{n} − ζ_{n}∥_{∞} goes to zero in probability, where ∥f∥_{∞} := sup_{0≤t≤1} |f(t)|. To this end let us introduce the random change of time θ_{n} defined as follows. When V_{n} > 0, θ_{n} is the map from [0, 1] onto [0, 1] which interpolates linearly between the points (k/n, V_{k}^{2}/V_{n}^{2}), k = 0, 1, …, n. When V_{n} = 0, we simply take θ_{n} = I, the identity on [0, 1]. With the usual convention S_{k}/V_{n} := 0 for V_{n} = 0, we always have

ζ_{n}(θ_{n}(t)) = V_{n}^{−1}ξ_{n}(t), 0 ≤ t ≤ 1.  (14)

Clearly for each t ∈ [0, 1],

|\frac{V_{[nt]}^{2}}{V_{n}^{2}} − θ_{n}(t)| ≤ max_{1≤k≤n} \frac{X_{k}^{2}}{V_{n}^{2}} .

It follows by (6) that

sup_{0≤t≤1}|\frac{V_{[nt]}^{2}}{V_{n}^{2}} − θ_{n}(t)| ⟶^{P} 0.

By Lemma 3.1 in [13],

sup_{0≤t≤1}|\frac{V_{[nt]}^{2}}{V_{n}^{2}} − t| ⟶^{P} 0.

So denoting by I the identity on [0, 1] we have

∥θ_{n} − I∥_{∞} ⟶^{P} 0.  (15)

Let ω(f; δ) := sup{|f(t) − f(s)|; |t − s| ≤ δ} denote the modulus of continuity of f ∈ C[0, 1]. Then recalling (14) we have

∥V_{n}^{−1}ξ_{n} − ζ_{n}∥_{∞} = sup_{0≤t≤1}|V_{n}^{−1}ξ_{n}(θ_{n}(t)) − ζ_{n}(θ_{n}(t))| ≤ ω(V_{n}^{−1}ξ_{n}; ∥θ_{n} − I∥_{∞}).

It follows that for any λ > 0 and 0 < δ ≤ 1,

P (∥V_{n}^{−1}ξ_{n} − ζ_{n}∥_{∞} ≥ λ) ≤ P (∥θ_{n} − I∥_{∞} > δ) + P (ω(V_{n}^{−1}ξ_{n}; δ) ≥ λ). (16)

Now since the Brownian motion has a version in C[0, 1], for each positive ε, we can find δ ∈ (0, 1] such that P (ω(W; δ) ≥ λ) < ε. As the functional ω is continuous on C[0, 1], it follows from Theorem 2.1 that

lim sup P (ω(V_{n}^{−1}ξ_{n}; δ) ≥ λ) ≤ P (ω(W; δ) ≥ λ).

Hence for n ≥ n_{1} we have P (ω(V_{n}^{−1}ξ_{n}; δ) ≥ λ) < 2ε. Having in mind (15) and (16) we see that the proof is complete.

3 Invariance principle in D[0, 1] under self-normalization

We now investigate the convergence in the Skorohod space of the step process V_{n}^{−1}W_{n} which jumps at deterministic times t = k/n and of the adaptive step process Z_{n} which jumps at random times t = V_{k}^{2}/V_{n}^{2}.

Theorem 3.1 The convergence

V_{n}^{−1}W_{n} ⟶^{𝒟} W  (17)

holds in the space D[0, 1] if and only if X_{1} ∈ DAN.

Theorem 3.2 The convergence

Z_{n} ⟶^{𝒟} W  (18)

holds in the space D[0, 1] if and only if X_{1} ∈ DAN.

Here again, the necessity of X_{1} ∈ DAN in both Theorems 3.1 and 3.2 follows from Giné, Götze and Mason [8].

Proof of Theorem 3.1. We have to prove tightness of the process W_{n} only. To this end we shall use Theorem 15.6 from Billingsley [4]. According to this theorem we have to find a nondecreasing continuous on [0, 1] function F such that

P (|W_{n}(t) − W_{n}(t_{1})| ≥ λ, |W_{n}(t_{2}) − W_{n}(t)| ≥ λ) ≤ λ^{−2γ}(F (t_{2}) − F (t_{1}))^{α}, (19)

for all 0 ≤ t_{1} ≤ t ≤ t_{2} and all n ≥ 1, where γ ≥ 0 and α > 1.

Let us denote by I_{λ}(t_{1}, t, t_{2}) the probability in the left hand side of (19). If t_{2} − t_{1} ≤ 1/n then either t, t_{1} ∈ [(k − 1)/n, k/n) or t, t_{2} ∈ [(k − 1)/n, k/n) for some k = 1, …, n. In both cases I_{λ}(t_{1}, t, t_{2}) = 0, so we may and do assume that t_{2} − t_{1} > 1/n. By Markov inequality we have

I_{λ}(t_{1}, t, t_{2}) ≤ λ^{−4}E V_{n}^{−4}(S_{[nt]} − S_{[nt_{1}]})^{2}(S_{[nt_{2}]} − S_{[nt]})^{2}.  (20)

To treat this bound let us remark that due to independence and identical distribution of the X_{i}’s, we have with d := ♯{i_{1}, i_{2}, i_{3}, i_{4}}

E \frac{X_{i_{1}}X_{i_{2}}X_{i_{3}}X_{i_{4}}}{V_{n}^{4}} = \{ \begin{matrix}E V_{n}^{−4}X_{1}^{2}X_{2}^{2} & when & d = 2; \\ E V_{n}^{−4}X_{1}X_{2}X_{3}^{2} & when & d = 3; \\ E V_{n}^{−4}X_{1}X_{2}X_{3}X_{4} & when & d = 4. \end{matrix}

After some combinatorics, this leads to the upper bound

I_{λ}(t_{1}, t, t_{2}) ≤ c\frac{(t_{2} − t_{1})^{2}}{λ^{4}} [n^{2}E \frac{X_{1}^{2}X_{2}^{2}}{V_{n}^{4}} + n^{3}E \frac{X_{1}X_{2}X_{3}^{2}}{V_{n}^{4}} + n^{4}E \frac{X_{1}X_{2}X_{3}X_{4}}{V_{n}^{4}} ].

It remains to check that the square brackets in the above estimate is bounded uniformly in n.

Since X_{1} ∈ DAN, we have by (1) and (2) the stochastic boundedness of the sequence (S_{n}/V_{n}). Now from Lemmas 2.1, 2.2 and 2.4 in Giné, Götze and Mason [8], it is easily seen that for some constant K depending only on the distribution of X_{1},

E V_{n}^{−4}X_{1}^{2}X_{2}^{2} ≤ Kn^{−2};
E V_{n}^{−4}X_{1}X_{2}X_{3}^{2} ≤ Kn^{−3};
E V_{n}^{−4}X_{1}X_{2}X_{3}X_{4} ≤ Kn^{−4}.

The proof is complete.

It is worth noticing that without the hypothesis X_{1} ∈ DAN, the stochastic boundedness of the sequence (S_{n}/V_{n}) is sufficient to obtain the tightness of the process V_{n}^{−1}W_{n}.

Proof of Theorem 3.2. We shall exploit Theorem 3.1 through the identity

Z_{n}(u) = V_{n}^{−1}W_{n}(θ_{n}^{−1}(u)), u ∈ [0, 1],

where

θ_{n}^{−1}(u) := sup{t ∈ [0, 1]; θ_{n}(t) = u}, u ∈ [0, 1],

is a generalized inverse of the random change of time introduced in the proof of Theorem 2.2. Observe that θ_{n}^{−1} is a non decreasing càdlàg function mapping [0, 1] into [0, 1]. By a classical result on the random changes of time in D[0, 1] (see e.g. Billingsley [4] p. 145 and Th. 4.4 ibidem), the problem is easily reduced to check the convergence in probability to zero of ∥θ_{n}^{−1} − I∥_{∞}. To this end we note that

∥θ_{n}^{−1} − I∥_{∞} = sup_{t∈[0,1]}|θ_{n}^{−1}(θ_{n}(t)) − θ_{n}(t)|
≤ sup_{t∈[0,1]}|θ_{n}^{−1}(θ_{n}(t)) − t| + ∥θ_{n} − I∥_{∞} .  (21)

Now for each t ∈ [0, 1] we have

0 ≤ θ_{n}^{−1}(θ_{n}(t)) − t ≤ \frac{L_{n}}{n} ,

where L_{n} is the discrete random variable equal to the maximum length of runs of consecutive occurences among the events {X_{i} = 0} (1 ≤ i ≤ n). Put p := P (X_{1} = 0). Since the case p = 0 is obvious and the hypothesis X_{1} ∈ DAN excludes the trivial case p = 1, we are left with the case 0 < p < 1 for which the elementary estimate

P (\frac{L_{n}}{n} > ε) ≤ np^{nε−1} = o(1),

gives the required convergence in probability of ∥θ_{n}^{−1} ∘ θ_{n} − I∥_{∞}. Together with (15) and (21), this makes the proof complete.

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