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An Inequality for Tail Probabilities of Martingales with Differences Bounded from One Side
Authors:V. Bentkus
Abstract:Let Mn=X1sdot sdot sdot +Xn be a martingale with differences Xk=MkMk–1bounded from above such that 
$$mathbb{P}{ X_k leqslant varepsilon _k } = 1$$
with some non-random positive epsik.Let the conditional variance xgr2k=E(X2k|X1,...,Xk–1) satisfy xgr2kles2k with probability one, where s2k are some non-random numbers. Write sgr2k=max{epsi2k,s2k} and sgr2=sgr21sdot sdot sdot +sgr2n. 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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