Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate Manuel L. Esqu´ ıvel *† Jo˜ ao Lita da Silva Jo˜ao Tiago Mexia Lu´ ıs Ramos May 5, 2011 This work is dedicated to Dinis D. F. Pestana, as a token of recognition for a lifelong enthusiasm for Statistics and for his never failing support of young mathematicians. Abstract Some asymptotic expansions non necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + μ n )) n∈ , where g is some smooth function, X is a random variable and (μ n ) n∈ is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace’s type integral, randomized by the sequence (μ n X) n∈ , X being a Gamma distributed random variable. 1 Introduction In this work we explore some asymptotic uniform approximations, for distribution func- tions, that do not require a central limit theorem as a starting point. The matter under investigation is well illustrated by the following example. Let X, Y be standard inde- pendent Gaussian variables, μ and ν parameters and α, β ≥ 3; the (exact) distribution of the random variable (μ + X ) α × (ν + Y ) β is not easily described but, for instance, performing a simulation experiment with varying and growing μ = ν parameter, we get 1