Polynomial expansions of density and distribution functions of scale mixtures

AbstractLet Y be an absolutely continuous random variable and W a nonnegative variable independent of Y. It is to be expected that when W is close to 1 in some sense, the distribution of the scale mixture YW will be close to Y. This notion has been investigated by a number of workers, who have provided bounds on the difference between the distribution functions of Y and YW. In this paper we examine the deeper problem of finding asymptotic expansions of the form P(YW ≤ x) = P(Y ≤ x) + Σn=1∞ E(Wr − 1)nGn(x), where r > 0 and the functions Gn do not depend on W. We approach the problem very generally, and then consider the normal and gamma distributions in greater detail. Our results are applied to obtain better uniform and nonuniform estimates of the difference between the distribution functions of Y and YW