We have focused on limit theorems for self-normalized sums so far. Sev- eral papers have recently discussed moment inequalities for self-normalized processes, de la Pena, Klass and Lai (2002) established very interesting exponential inequalities for general self-normalized processes.
Theorem 8.1. [de la Pena, Klass and Lai (2002)]. Let S and V be two variables with V > 0 such that
£ e x p { A S - — V2}<1 (28)
for all A £ R. Then for all y > 0,
v S2
E . y exp{ ——5 sr-} < 1. (29)
y/V*+y2 *X2(V2 + y2)s- v ' Consequently, if EV > 0, then
E e xK4( ^ - T ( E V T ) ) - ^
and
xS for x > 0. Moreover, for all p > 0,
( xS \
==) < V2 exp(x2) JV*A-(EV\*J ~
EJ , |51 =Y < 2p+2Pr(p/2).
•^/V2 + {EV)2'
Condition (28) is satisfied for a large classes of random variables (5, V).
Important examples include: (i) S = WT, V = y/T, where Wt is a standard Brownian motion, and T is a stopping time with T < oo a.s.; (ii) S = Mt,
V — ^ / ( Mt) , where Mt is a continuous square-integrable martingale w i t h M0 = 0; (iii) S = YLl=i ^t> ^ = \ / 2 r = i ^?> w n e r e {di\ is a sequence of variables a d a p t e d to an increasing sequence of cr-fields {Fj} and t h e dj's are conditionally symmetric. T h e y also developed maximal inequalities a n d iterated logarithm bounds for self-normalized martingales.
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SELF-NORMALIZED SUMS*
B I N G - Y I JING*
Department of Mathematics,
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
E-mail: majing@ust.hk
Limit theorems for sums and self-normalized sums of independent random variables occupy a special place in probability and mathematical statistics. First, they are t h e simplest statistics to study mathematically so that many interesting results can be derived. Secondly, their studies can shed light on other classes of statistics such as the function of the sum of independent random variables, (/-statistics, L- statistics and symmetric statistics. Finally, they are also very useful in practice themselves, e.g., the self-normalized sums are closely related to the commonly used Student t-statistics in inference. In this paper, we shall briefly review some of the results for sums and self-normalized sums of independent random variables.
Particular attention will be given to t h e similarities and differences of the two cases.
1. Introduction
Let Xi, • • • , Xn be a sequence of independent non-degenerate random vari- ables such that EXj = 0 and var(Xj) = CT| < oo, j = 1, ...,n. Let
Sn = ±Xj, B>n = ±al V* = ±Xl Wl = ±X*-l-Sl
3 = 1 j=l j = l j=l
Clearly, Bn is the variance of Sn; V% and W% are two different estimates of B2n. Define
Tin = Sn/Bn, T<2n — Sn/Vn, T3n = Sn/Wn,
"This work is supported by etc, etc.
tWork partially supported by grant 2-4570.5 of the Swiss National Science Foundation.
69
which are, respectively, the standardized, self-normalized, and studentized sums. Also define their respective distribution functions by
Fln(x)=P(Sn/Bn<x), F2n(x)=P(Sn/Vn<x), F3n(x)=P(SJWn<x).
Note that the Student t-statistic T$n and the self-normalized sums Tin have a 1-1 correspondence through the relationship
,1 / 2 Tan = Tin n-T2n/
Therefore, it suffices to study T\n and Tin below.
In this paper, we shall summarize some of the main results associated with Tin and Tin- In particular, we shall discuss the asymptotic normality, Berry-Esseen bounds, Edgeworth expansions, small and moderate to large deviations and saddlepoint approximations. Also we shall give some recent results on LIL for partial sums and increments of partial sums for self- normalized sums.