Pair correlations and $U$-statistics for independent and weakly dependent random variables

Pair Correlations and U-statistics for Independent and Weakly Dependent Random Variables

We prove a Glivenko-Cantelli type strong law of large numbers for the pair correlation of independent random variables. Except for a few powers of logarithms the results obtained are sharp. Similar estimates hold for the pair correlation of lacunary sequences {nkω} mod 1.

Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables

Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...

On the Asymptotic Independent Representations for Sums of Some Weakly Dependent Random Variables

Let, for each n ∈ N, (Xi,n)05i5n be a triangular array of stationary, centered, square integrable and associated real valued random variables satisfying the weakly dependence condition lim N→N0 lim sup n→+∞ n n X r=N Cov (X0,n, Xr,n) = 0; where N0 is either infinite or the first positive integer N for which the limit of the sum n Pn r=N Cov (X0,n, Xr,n) vanishes as n goes to infinity. The purpo...

Central and Local Limit Theories in Markov Dependent Random Variables

Strong Laws for Weighted Sums of Negative Dependent Random Variables

In this paper, we discuss strong laws for weighted sums of pairwise negatively dependent random variables. The results on i.i.d case of Soo Hak Sung [9] are generalized and extended.