# Almost sure invariance principles for partial sums of weakly - download pdf or read online

By Walter Philipp

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**Additional info for Almost sure invariance principles for partial sums of weakly dependent random variables**

**Sample text**

In Esary, Proschan, and Walkup (1967), it is shown that S and T are positively quadrant dependent if and only if C o v (f(S),g(T)) >_0 36 for all nondecreasing measurable functions f, g : R --+ R, whenever this covariance is defined. Thus S and T are positively quadrant dependent if and only if {S, T} is quasi-associated. Since our first theorem is from Khoshnevisan and Lewis (1997), its proof will be omitted. 1. T h e o r e m . Let {Zk : k >_ 1} be a sequence of pairwise positively quadrant depen- dent random variables with bounded second moments.

CsSrgS, S. (1974 b), On weak convergence of the empirical process with random sample size, Acta Sci. Math. Szeged 36, 17-25. 29 CsSrgS, S. (1997), Rdnyi-mixing in the generalized arc-sine law, Ann. Univ. Mariae Curie-Sktodowska, Sectio A (Mathematica) 51, 15-25. CsSrgS, S. (1998), Rdnyi-mixing of occupation times, in this volume. T. I. Resnick (1977), Weak convergence with random indices, Stochastic Process Appl. 5, 213-220. Guiasu, S. (1971), On the asymptotic distribution of the sequences of random variables with random indices, Ann.

T,_~-6)/BN,,) <_ x] i=l M [(A - 5)B. < BN. ] A [ti-1 < A <_ ti] A A). On the other hand if t~_l < A _< ti and (18) (A - 5)B~ < BN. (t,_,_~)/BN. (t,_,-~)/(ti-1 - 5)Bn. Thus, since e > 0, 5 > 0 and rl > 0 can be chosen a r b i t r a r y small, 5 < ~, (3) follows from (4), (2), ( 1 7 ) a n d (18). Now assume A - const. > 0 with probability one. ) < x] N [(A - r < BN. ] ) - P (IBN. /BN. <_ X) <_ P (IBN. CA_e))/(Sm. ) <_ x] M [(A - e)Sn < Sg,~ < (A + e)S,~] ). , < (A + r #. , >_ BM,~(A+e)/(A + e)B~ , 1, as n ~ oo.