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

In this note, 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.


INTRODUCTION
Let X, X 1 , X 2 , . . .be i.i.d.random variables with mean zero and variance one, and set Also let log x = ln(x ∨ e), log 2 x = log(log x).Then by the so-called Chung's law of the iterated logarithm we have lim inf n→∞ log 2 n/n M n = π/ √ 8 a.s. (1.1) 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, EX 2 = 1 and (1.2)
Then for any b > −1, we have Throughout this note, let C denote a positive constant, whose values can differ in different places.

PROOF
PROOF OF THEOREM 1.1.It is enough to prove the result for eventually non-decreasing function Observe that by (2.1) (2.2) and by Theorem 1 of Einmahl (1993), we have Now by Lemma 2.2 below, 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.

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Applying Kolmogorov's LIL and EX 2 = 1, we have lim sup it easily follows from above inequalities that (2.9) Hence observe that on account of (2.9) it is enough to show that (2.10) Thus, in order to prove (2.10), it suffices to show that Using the independence and Lemma 2.1, it is clear that for some constant η > 0 and large enough k, By Lemma 2.1, there exists a positive constant η such that Since EX = 0 and EX 2 = 1, there exists a positive integer n 0 such that for all n ≥ n 0 E X 2 n1 ≥ 3 4 and E Xn1 ≤ 1 4 .
Hence using the Bernstein inequality, there exists a positive constant β < 1/3000 such that Finally, by EX 2 = 1, we have Thus, (2.12) holds true.