Self-normalized deviation inequalities with application to t-statistic

Let (ξi)i≥1 be a sequence of independent and symmetric random variables. We obtain some upper bounds on tail probabilities of self-normalized deviations P(max1≤k≤n∑i=1kξi/(∑i=1n∣ξi∣β)1/β≥x) for x>0 and β>1. Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's t-statistic is also given.