Ph. Barbe

On Sharp Large Deviations for Sums of Random Vectors and Multidimensional Laplace Approximation

Let

Large-deviation probability and the local dimension of sets

X, X_i,i\ge 1

Last passage time for the empirical mean of some mixing processes

, be a sequence of independent and identically distributed random vectors in

Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution

{\bf R}^d

Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions

. Consider the partial sum

Blowing number of a distribution for a statistics and loyal estimators

S_n:=X_1+\cdots +X_n

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

. Under some regularity conditions on the distribution of X, we obtain an asymptotic formula for

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

P\{S_n\in nA\}

Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

, where A is an arbitrary Borel set. Several corollaries follow, one of which asserts that, under the same regularity conditions, for any Borel set A,

Second-order expansion for the maximum of some stationary Gaussian sequences

\lim_{n\to\infty}n^{-1}\log P\{S_n\in nA\} =-I(A)