A Gambling Game Arising in the Analysis of Adaptive Randomized Rounding

Abstract

Let y be a positive real number and let {X _i } be an infinite sequence of Bernoulli random variables with the following property: in every realization of the random variables, $\sum_{i=1}^{\infty} E[X_i|X_1,X_2,\cdots, X_{i-1}] \leq y$. We specify a function F(x,y) such that, for every positive integer x and every positive real y, $P(\sum_{i=1}^{\infty} X_i \geq x) \leq F(x,y)$; moreover, for every x and y, F(x,y) is the best possible upper bound. We give an interpretation of this stochastic process as a gambling game, characterize optimal play in this game, and explain how our results can be applied to the analysis of multi-stage randomized rounding algorithms, giving stronger results than can be obtained using the traditional Hoeffding bounds and martingale tail inequalities.