Maximal Inequalities for Dependent Random Variables
Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X1, X2, … be random variables with par
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Abstract Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X1 ; X2 ; : : : be random variables with partial sums Sk D X1 C C Xk . Then a maximal inequality gives conditions ensuring that the maximal partial sum Mn D max1in Si is of the same order as the last sum Sn . In the literature there exist large number of maximal inequalities if X1 ; X2 ; : : : are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X1 ; X2 ; : : : having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions. Keywords Demi-martingales • Integral orderings • Mixing conditions • Negative and positive correlation
Mathematics Subject Classification (2010). Primary 60E15; Secondary 60F05
1 Introduction Q FQ ; P/ Q denote two fixed probability Throughout this paper, we let .; F ; P/ and .; spaces. We let R D .1; 1/ denote the real line and we let RC D Œ0; 1/ denote the non-negative real line. We let N0 D f0; 1; 2; : : :g denote the set of all nonnegative integers, we let N D f1; 2; : : :g denote the set of all positive integers and we define n D f.i; j/ 2 N02 j j i ng ; n D f.i; j/ 2 N02 j 0 j i ng 8 n 2 N0 ; r D f.i; k; j/ 2 N03 j i < k jg :
J. Hoffmann-Jørgensen () Department of Mathematics, University of Aarhus, Aarhus, Denmark e-mail: [email protected] © Springer International Publishing Switzerland 2016 C. Houdré et al. (eds.), High Dimensional Probability VII, Progress in Probability 71, DOI 10.1007/978-3-319-40519-3_4
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Let X1 ; X2 ; : : : be a sequence of random variables and let us consider the partial sums and the maximal partial sums : Si;i D Mi;i D M i;i D 0 and Si;j D
P
Xk ; Mi;j D max Si;k ; M i;j D max jSi;k j 8 .i; j/ 2 1 : ikj
i t/ E 1fMi;j >tg '.Si;j / 8 t 2 R :
(1.10)
If x 2 R, we let bxc denote the largest integer x and we let dxe denote the smallest integer x. We let R D Œ1; 1 denote the extended real line and I shall use the following extension of the arithmetic on the real line R D .1; 1/: x C 1 WD 1 8 1 x 1 ; x C .1/ WD 1 8 1 x < 1 ; 0 .˙1/ WD 0 ; x .˙1/ WD ˙1 ; .x/ .˙1/ WD 1 8 0 < x 1 ; 1 0
D log 1 D e1 WD 1 ;
1 ˙1
D e1 WD 0 ;
x y
WD x
1 y
; x0 D 1 8 x; y 2 R ;
and I shall Puse the standard conventions inf ; D min ; WD 1, sup ; D max ; WD 1 and k2; ak WD 0. If V is a real vector space, we say that W V ! R is sub-additive if .x C y/ .x/ C . y/ for all x; y 2 V. Let k 1 be an integer. Then we let B k denote the Borel -algebra on Rk and we let denote the coordinate-wise ordering on Rk ; that is .x1 ; : : : ; xk / . y1 ; : : : ; yk / if and only if xi yi for all i D 1; : : : ; k. If D Rk and F W D ! Rm is a function, we say that F is increasing if F.x/ F. y/ for all x; y 2 D with x y. If u D .u1 ; : : : ; uk / and v
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