Self-normalized deviation inequalities with application to $t$-statistics

Let $(\xi_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$\mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |\xi_i|\big/ \big(\sum_{i=1}^{n} |\xi_i|^\beta \big)^{1/\beta} \geq x \Big)$$ for $x>0$ and $\beta >1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.