Limiting distribution of sums of nonnegative stationary random variables
The paper presents a limit theorem on the convergence of the distribution of the sum in an array of nonnegative random variables. The array is stationary in the following sense: if the array is represented as \(\{X_{n,j}\), \(1\leq j\leq n<\infty \}\), then, for each n, the random variables \(X_{n,j}\), \(j=1,...,n\), form a stationary (though finite) sequence. The hypothesis of this theorem includes certain kinds of mixing conditions which provide enough independence to give results similar to those for independent summands. The novelty of this work is the introduction of a new mixing condition having more flexibility in certain applications. The main theorem. Assume the following two conditions: (i) There exists a nonincreasing function H(x), \(x\geq 0\), such that \(n\int^{x}_{0}\min (y,1)dP[X_{n,0}>y]\to \int^{x}_{0}\min (y,1)dH(y)\), in the sense of complete convergence on \(x>0\). (ii) There exists a sequence \(\{\gamma_ n\}\) of positive integers such that for every \(c>0\), \(n\sum_{1\leq j\leq \gamma_ n}P[X_{n,0}>c\), \(X_{n,j}>c]\to 0\) and, for every \(q>0\), \(P[X_{n,0}>x| X_{n,j}\), \(j\leq -q\gamma_ n]/P[X_{n,0}>x]\to^{P}1\) (P - in probability). Then \(X_{n,1}+...+X_{n,n}\) has, for \(n\to \infty\), a limiting distribution with the Laplace-Stieltjes transform \(\phi (s)=\exp [\int^{\infty}_{0}(1-e^{-sx})dH(x)].\) There are also given applications to functions defined on a stationary sequence and to the limiting distribution of the maximum in a stationary sequence.
distribution of the maximum in a stationary sequence
scientific article; zbMATH DE number 3878060