Exponential bounds via Stein’s method and exchangeable pairs

Abstract

We apply Stein's method to obtain a non-uniform exponential bound for normal approximations for certain types of random variables. In particular, we establish the bound for a random variable such that its exchangeable pair coupling exists and the distance between the pair is bounded. Using our result, we obtain better bounds for a wide range of applications, such as sums of bounded independent random variables, the combinatorial central limit theorem, and the numbers of descents and inversions of a permutation.

An improvement of a nonuniform bound for unbounded exchangeable pairs

“…To obtain the bound in (9), we apply the technique in Sumritnorrapong, Neammanee, and Suntornchost [8] to each of the three terms as follows.…”

“…The exchangeable-pair approach has been intensively used as a main tool in constructing bounds of distribution approximation. For instance, Rinott and Rotar [6], Shao and Su [7], and Sumritnorrapong, Neammanee, and Suntornchost [8] applied the exchangeable-pair approach to obtain uniform bounds for a normal approximation of a random variable where the the exchangeable pair exists and the difference between the exchangeable pairs is bounded. That is, |W -W | ≤ A, for a positive constant A.…”

“…One application is the extension of the uniform bound for the quadratic forms in Shao and Zhang [9] to a nonuniform bound. In this paper, with a different approach, we extend the technique in Sumritnorrapong, Neammanee, and Suntornchost [8] to propose a nonuniform bound of the normal approximation when the boundedness of the distance between the exchangeable pair is not required. Moreover, we apply our obtained bound to improve the error bound of the normal approximation in two well-known applications that are the independence test and the quadratic forms.…”