Approximating sums of products of dependent random variables

Abstract

Stochastic approximation of a given time series $\left\{{\sum }_{j=1}^k{X}_j{Y}_j\right\}$ by a linear combination of simpler sequences $\left\{{\sum }_{j=1}^k{X}_j\right\}$ and $\left\{{\sum }_{j=1}^k{Y}_j\right\}$ is treated uniformly over $k\in \{1,\dots ,n\}$. A maximal inequality is proven in order to find a sharp bound on Value-at-Risk of ${max}_{1\le k\le n}\left|{\sum }_{j=1}^k{X}_j{Y}_j\right|$.