An Inequality for Tail Probabilities of Martingales with Differences Bounded from One Side

V. Bentkus ()

Journal of Theoretical Probability, 2003, vol. 16, issue 1, 161-173

Abstract: Abstract Let M n =X 1+ ⋅ ⋅ ⋅ +X n be a martingale with differences X k =M k −M k−1bounded from above such that $$\mathbb{P}\{ X_k \leqslant \varepsilon _k \} = 1$$ with some non-random positive ε k .Let the conditional variance ξ 2 k =E(X 2 k |X 1,...,X k−1) satisfy ξ 2 k ≤s 2 k with probability one, where s 2 k are some non-random numbers. Write σ 2 k =max{ε 2 k ,s 2 k } and σ 2=σ 2 1+ ⋅ ⋅ ⋅ +σ 2 n . We prove the inequality $$\mathbb{P}\{ M_n \geqslant x\} \leqslant \min \{ \exp \{ - x^2 /(2\sigma ^2 )\} ,c_0 (1 - \Phi (x/\sigma ))\} $$ with a constant $$c_0 = 1/(1 - \Phi (\sqrt 3 )) \leqslant 25$$ .

Keywords: Probabilities of large deviations; martingale; bounds for tail probabilities; inequalities; bounded differences and random variables; measure concentration phenomena; product spaces; Lipschitz functions; Hoeffding's inequalities; Azuma's inequality (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1022234622381

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