On the other law of the iterated logarithm for self-normalized sums

MATHEMATICAL SCIENCES

On the other law of the iterated logarithm for self-normalized sums

Guang-Hui Cai

Department of Mathematics and Statistics, Zhejiang Gongshang University, Hangzhou 310035, P.R. China

ABSTRACT

Inthisnote, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.

Key words: Chung's integral test, self-normalized sums, convergence rate.

RESUMO

Nesta nota, obtemos um teste integral de Chung para somas auto-normalizadas de variáveis aleatórias i.i.d. (independentes e identicamente distribuídas). Além disso, obtemos uma taxa de convergência da lei de Chung do logaritmo iterado para somas auto-normalizadas.

Palavras-chave: teste integral de Chung, somas auto-normalizadas, taxa de convergência.

1 INTRODUCTION

Let X, X₁, X₂,... be i.i.d. random variables with mean zero and variance one, and set

Also let logx = ln(x ∨ e), log₂x = log(logx). Then by the so-called Chung's law of the iterated logarithm we have

This result was first proved by Chung (1948) under E|X|³ < ∞, and by Jain and Pruitt (1975) under the sole assumption of a finite second moment. Einmahl (1989) obtained the Darling Erdös theorem for sums of i.i.d. random variables. Griffin and Kuelbs (1989) got Self-normalized laws of the iterated logarithm. Griffin and Kuelbs (1991) obtained some extensions of the laws of the iterated logarithm via self-normalized.Lin (1996) got a self-normalized Chung-type law of iterated logarithm. Einmahl (1993) obtained the following integral test refining (1.1) under the minimal conditions.

THEOREM A. Let {X, X_n; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX² = 1 and

Then for any eventually non-decreasing function Ø:[1, ∞) → (0, ∞),

Einmahl (1993) showed that if (1.2) is not true, Theorem A is false. We thus see that condition (1.2) is sharp. However, if we use V_n to replace , we can eliminate the condition (1.2) in Theorem A. Explicitly, we get the following theorem.

THEOREM 1.1. Let {X, X_n; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX² = 1. Then for any eventually non-decreasing function Ø:[1, ∞) → (0, ∞),

Our next theorem gives a result on a convergence rate of (1.1).

THEOREM 1.2. Let {X, X_n; n > 1} be a sequence of i.i.d. random variables with EX = 0, EX² = 1. Then for any b > -1, we have

Throughout this note, let C denote a positive constant, whose values can differ in different places.

2 PROOF

PROOF OF THEOREM 1.1. It is enough to prove the result for eventually non-decreasing function Ø:[1, ∞) → (0, ∞) satisfying

(See Einmahl 1993). Let

Observe that by (2.1)

where Ψ(t) = Ø(t)³/(1+Ø(t)²), t > 1, and similarly,

where Ψ'(t) = Ø(t)³/(Ø(t)² -1), t > 1. It is easily checked that J(Ø) < ∞ implies J(Ψ) < ∞ and J(Ø) = ∞ implies J(Ψ^') = ∞, and by Theorem 1 of Einmahl (1993), we have

J(Ψ) < ∞ ⇒ P (M_n < B_n / Ψ(n) i.o. = 0

and

J(Ψ') = ∞ ⇒ P (M_n < B_n / Ψ'(n) i.o. = 0

Now by Lemma 2.2 below,

P(M_n< V_n / Ø(n), Δ_n>(log₂n)^-3/2 i.o.) = 0

and

P(M_n< B_n / Ø(n), Δ_n>(log₂n)^-3/2 i.o.) = 0.

From these equations and (2.2), (2.3), hence we see that Theorem 1.1 holds true.

We now present two lemmas used in the main proof of Theorem 1.1.

LEMMA 2.1. For any x > 0 there exist positive constants η = η(x) and A = A(x) such that

PROOF. See the Lemma 2(b) of Einmahl (1993).

LEMMA 2.2. We have

and

PROOF. Let

and

First using EX² = 1, we have

and

Thus, it follows by applying Corollary 3.1 of Lin et al. (1999, P.95) and Borel-Cantelli lemma that

Using strong law of large numbers and Hartman-Wintner LIL, we have

Thus, by EX² = 1, we obtain that for large n,

Recalling that

< n, n > 1 and lim_{n→ ∞} /n = 1 a.s., in order to prove (2.4) and (2.5), by (2.1), (2.6) and (2.7), it suffices to show that

Now, set m(n):= [n/(log₂n)⁹], n > 1. By EX² = 1, we have

Applying Kolmogorov's LIL and EX² = 1, we have

it easily follows from above inequalities that

Hence observe that on account of (2.9) it is enough to show that

Let n_k = 2^k and m_k = [2^k/(logk)¹⁰], k > 0, for large enough k,

Thus, in order to prove (2.10), it suffices to show that

Let

Notice that

M_{nk-1, j}<M_nk-1+|X_j| < 3 M_nk-1, 1 < j < n_k-1.

Using the independence and Lemma 2.1, it is clear that for some constant η > 0 and large enough k,

Finally, By Lemma 4 of Einmahl (1993), we have

and hence we obtain (2.11) from the Borel-Cantelli lemma.

PROOF OF THEOREM 1.2. For each n > 1 and 1 < i < n, we have

By EX² = 1, it is easy to show that < n. Hence for some < δ < 1 and any ε > 0

In order to prove (1.5), it suffices to show that for any b > -1

By Lemma 2.1, there exists a positive constant η such that

Since EX = 0 and EX² = 1, there exists a positive integer n₀ such that for all n > n₀

Hence using the Bernstein inequality, there exists a positive constant β < 1/3000 such that

Finally, by EX² = 1, we have

Thus, (2.12) holds true.

ACKNOWLEDGMENTS

This paper was supported by National Natural Science Foundation of China and Youth Talent Foundation of Zhejiang Gongshang University (Q07-07).

Manuscript received on March 17, 2008; accepted for publication on May 15, 2008; presented by ARON SIMIS

AMS Classification: 60F15, 62F05

E-mail: [e-mail]

• CHUNG KL. 1948. On the maximal partial sums of sequences of independent random variables. Trans Amer Math Soc 4: 205233.
• EINMAHL U. 1989. The Darling Erdös theorem for sums of i.i.d. random variables. Probab Theory Relat Fields 82: 241257.
• EINMAHL U. 1993. On the other law of the iterated logarithm. Probab Theory Relat Fields 96: 97106.
• GRIFFIN PS AND KUELBS JD. 1989. Self-normalized laws of the iterated logarithm. Ann Probab 17: 15711601.
• GRIFFIN PS AND KUELBS JD. 1991. Some extensions of the laws of the iterated logarithm via self-normalized. Ann Probab 19: 380395.
• JAIN NC AND PRUITT WE. 1975. The other law of the iterated logarithm. Ann Probab 3: 10461049.
• LIN ZY. 1996. A self-normalized Chung-type law of iterated logarithm. Theory Probab Appl 41: 791798.
• LIN ZY, LU CR AND SU AG. 1999. Limit Theoretical Basis of Probability. Hight Education Press, Beijing.

Publication Dates

• Publication in this collection
  28 Aug 2008
• Date of issue
  Sept 2008

History

• Accepted
  15 May 2008
• Received
  17 Mar 2008