Bounding Probability of Small Deviation: A Fourth Moment Approach

In this paper we study the problem of bounding the value of the probability distribution function of a random variable X at E[X] + a where a is a small quantity in comparison with E[X], by means of the second and the fourth moments of X. In this particular context, many classical inequalities yield only trivial bounds. By studying the primal-dual moments-generating conic optimization problems, we obtain upper bounds for Prob {X ≥ E[X] + a}, Prob {X ≥ 0}, and Prob {X ≥ a} respectively, where we assume the knowledge of the first, second and fourth moments of X. These bounds are proved to be tightest possible. As application, we demonstrate that the new probability bounds lead to a substantial sharpening and simplification of a recent result and its analysis by Feige ([7], 2006); also, they lead to new properties of the distribution of the cut values for the max-cut problem. We expect the new probability bounds to be useful in many other applications.