Exponential inequalities for self-normalized processes with applications

Victor H. De La Peña, Guodong Pang

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 − 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.

Original languageEnglish (US)
Pages (from-to)372-381
Number of pages10
JournalElectronic Communications in Probability
Volume14
DOIs
StatePublished - Jan 1 2009

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Exponential Inequality
Martingale Difference Sequence
Exponential Bound
Tail Probability
Stopping Time
Lp-norm
Hypothesis Testing
Martingale
Random variable

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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

In: Electronic Communications in Probability, Vol. 14, 01.01.2009, p. 372-381.

Research output: Contribution to journalArticle

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AU - Pang, Guodong

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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.

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