Fe b 20 07 Bahadur representation of sample quantiles for functional of Gaussian dependent sequences under a minimal assumption

We consider the problem of obtaining a Bahadur representation of sample quantiles in a certain dependence context. Before stating in what a Bahadur representation consists, let us specify some general notation. Given some random variable Y , F(·) = FY (·) is referred as the cumulative distribution function of Y , ξ(p) = ξY (p) for some 0 < p < 1 as the quantile of order p. If F(·) is absolutely continuous with respect to Lebesgue measure, the probability density function is denoted by f(·) = fY (·). Based on the observation of a vector Y = (Y (1), . . . , Y (n)) of n random variables distributed as Y , the sample cumulative distribution function and the sample quantile of order p are respectively denoted by F̂Y (·;Y ) and ξ̂Y (p;Y ) or simply by F̂ (·;Y ) and ξ̂ (p;Y ). Let Y = (Y (1), . . . , Y (n)) a vector of n i.i.d. random variables such that F ′′(ξ(p)) exists and is bounded in a neighborhood of ξ(p) and such that F ′(ξ(p)) > 0, Bahadur proved that as n → +∞,