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

Authors:

V. Bentkus

Abstract:

Let M_n=X₁+ sdot

+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 _k.Let the conditional variance ²_k=E(X²_k|X₁,...,X_k–1) satisfy ²_k

s²_k with probability one, where s²_k are some non-random numbers. Write ²_k=max{²_k,s²_k} and ²=²₁+ sdot

+²_n. We prove the inequality

$mathbb{P}{ M_n geqslant x} leqslant min { exp { - x^2 /(2sigma ^2 )} ,c_0 (1 - Phi (x/sigma ))}$

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

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