Truncation in average and worst case settings for special classes of [formula omitted]-variate functions

Abstract

The paper considers truncation errors for functions of the form f(x1,x2,…)=g(∑j=1∞xjξj), i.e., errors of approximating f by fk(x1,…,xk)=g(∑j=1kxjξj), where the numbers ξj converge to zero sufficiently fast and xj’s are i.i.d. random variables. As explained in the introduction, functions f of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides bounds on truncation errors in both the average and worst case settings. The bounds are sharp in two out of three cases that we consider. In the former case, the functions g are from a Hilbert space G endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions g are from a reproducing kernel Hilbert space, or a space of functions satisfying a Hölder condition.