On the asymptotic of likelihood ratios for self-normalized large deviations

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Motivated by multiple statistical hypothesis testing, we obtain the limit of likelihood ratio of large deviations for self-normalized random variables, specifically, the ratio of P(√n(X +d/n) ≥ xₙ V) to P(√n X ≥ xₙ V), as n, where X and V are the sample mean and standard deviation of iid X₁,..., Xₙ, respectively, d>0 is a constant and xₙ. We show that the limit can have a simple form eᵈ/ᶻ⁰, where z₀ is the unique maximizer of z f(x) with f the density of Xᵢ. The result is applied to derive the minimum sample size per test in order to control the error rate of multiple testing at a target level, when real signals are different from noise signals only by a small shift.