On probabilities of large deviations of sums of dependent random variables. (English. Russian original) Zbl 0627.60033

Amosova, N. N.

Theory Probab. Math. Stat. 32, 1-7 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 3-9 (1985).

If \(\{X_ n,n\geq 1\}\) is a sequence of random variables, then they are m-dependent or m-orthogonal whenever \(X_ n\) and \(X_ k\) are independent or orthogonal for \(| n-k| \geq k+1\). Let \(S_ n=\sum^{n}_{i=1}X_ i\). For such a sequence the limit behavior of \[ P(S_ n\geq \epsilon t_ n),\quad P(\max_{1\leq k\leq n}S_ k\geq \epsilon t_ n),\quad and\quad P(\sup_{k\geq n}S_ k/t_ k\geq \epsilon) \] is studied with \(\epsilon >0\) and \(t_ n>0\) as constants. “Good” sufficient conditions are given in order that the probabilities tend to zero at a prescribed rate as \(n\to \infty\). These results generalize some of the earlier authors’ work on the problem.

Reviewer: M.M.Rao

MSC:

60F10	Large deviations
60G50	Sums of independent random variables; random walks

Keywords:

m-dependent random variables; m-orthogonal random variables