Exponential deficiency of convolutions of densities

e〈x,tu〉 p(x) dx < ∞ for some linear functional u and all t ∈ (0, 1), then, for each t ∈ (0, 1) and all large enough n, the n-fold convolution of the ttilted density p̃t(x) := e〈x,tu〉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. AMS 2000 subject classifications: Primary 60E05, 60E10; secondary 60F10, 62E20, 60E15.