Bernstein-type large deviations inequalities for partial sums of strong mixing processes

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Abstract
By using an approximation of partial sums of aprocess by independent r.v.'s we obtain large deviation inequalities. In particular we show that if is a centered real valued process, if there exist constants m and M such that and if 1≥p n ≥n/2 then, for every ϵ>0, we have where (α(p), p ≥ 1) denote ihe strong mixing coefficients of (Xt). This inequality is simpler and more explicit than those which are actually available in the literature. Similarly we obtain an inequality for processes satisfying Cramer's conditions. Applications to the rate of convergence in the law of large numbers and to nonparametric regression with dependent errors are given.