Exponential deficiency of convolutions of densities

TL;DR

This paper proves that convolutions of certain tilted densities become bounded as the number of convolutions increases, which is useful for saddle-point approximations and has implications for understanding the behavior of sums of random variables.

Contribution

It establishes a general non-i.i.d. result showing the exponential deficiency of convolutions of bounded densities under linear tilts, with an optimality property.

Findings

Convolutions of tilted densities become bounded for large enough n.

The result applies to non-i.i.d. settings and is optimal.

Useful for saddle-point approximation techniques.

Abstract

If a probability density p(\x) (\x\in\R^k) is bounded and R(t) := \int \exp(t\ell(\x)) \d\x < \infty for some linear functional \ell and all t\in(0,1), then, for each t\in(0,1) and all large enough n, the n-fold convolution of the t-tilted density p_t(\x) := \exp(t\ell(\x)) p(\x)/R(t) is bounded. This is a corollary of a general, "non-i.i.d." result, which is also shown to enjoy a certain optimality property. Such results are useful for saddle-point approximations.