## Abstract

We prove the following exponential inequality for a pair of random variables (A, B) with B > 0 satisfying the canonical assumption, E[exp(λA − λ^{2}/2 B^{2})] ≤ 1 for λ ∈ ℝ, p(|A|/√ 2q−1/q(B^{2} + (E[|A|^{p}])^{2/p}) ≥ x) ≤ (q/2q − 1)^{q/2q−1} x^{−q/2q−1} e^{−x2/2} for x > 0, where 1/p + 1/q = 1 and p ≥ 1. Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the Lp-norm (p ≥ 1) of A (in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in [0,1]d (d ≥ 2), connected to the CLT.

Original language | English (US) |
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Pages (from-to) | 372-381 |

Number of pages | 10 |

Journal | Electronic Communications in Probability |

Volume | 14 |

DOIs | |

State | Published - Jan 1 2009 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty