Large deviations viewpoints for a heavy-tailed β-mixing sequence

Let (Xn;n≥1) be a stationary sequence of non-negative random variables with heavy tails. Under mixing conditions, we study logarithmic asymptotics for the distributions of the partial sums sn=X1 X2 … Xn. We obtain the crude estimates P(Sn>nx)≈n-αx 1 for appropriate values of x, where a is a specific parameter. The related conjecture proposed by Gantert is investigated. As a by-product, the so-called supremum large deviations principle is also studied.