### 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) |
---|---|

Pages (from-to) | 372-381 |

Number of pages | 10 |

Journal | Electronic Communications in Probability |

Volume | 14 |

DOIs | |

State | Published - Jan 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Communications in Probability*,

*14*, 372-381. https://doi.org/10.1214/ECP.v14-1490

}

*Electronic Communications in Probability*, vol. 14, pp. 372-381. https://doi.org/10.1214/ECP.v14-1490

**Exponential inequalities for self-normalized processes with applications.** / De La Peña, Victor H.; Pang, Guodong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Exponential inequalities for self-normalized processes with applications

AU - De La Peña, Victor H.

AU - Pang, Guodong

PY - 2009/1/1

Y1 - 2009/1/1

N2 - We prove the following exponential inequality for a pair of random variables (A, B) with B > 0 satisfying the canonical assumption, E[exp(λA − λ2/2 B2)] ≤ 1 for λ ∈ ℝ, p(|A|/√ 2q−1/q(B2 + (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.

AB - We prove the following exponential inequality for a pair of random variables (A, B) with B > 0 satisfying the canonical assumption, E[exp(λA − λ2/2 B2)] ≤ 1 for λ ∈ ℝ, p(|A|/√ 2q−1/q(B2 + (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.

UR - http://www.scopus.com/inward/record.url?scp=70349448354&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349448354&partnerID=8YFLogxK

U2 - 10.1214/ECP.v14-1490

DO - 10.1214/ECP.v14-1490

M3 - Article

AN - SCOPUS:70349448354

VL - 14

SP - 372

EP - 381

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

ER -