TY - JOUR
T1 - Bounding probability of small deviation
T2 - A fourth moment approach
AU - He, Simai
AU - Zhang, Jiawei
AU - Zhang, Shuzhong
PY - 2010/2
Y1 - 2010/2
N2 - In this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount (relative to the expected value), by means of the second and the fourth (central) moments of the random variable. In this particular context, many classical inequalities yield only trivial bounds. We obtain tight upper bounds by studying the primal-dual moments-generating conic optimization problems. As an application, we demonstrate that given the new probability bounds, a substantial sharpening and simplification of a recent result and its analysis by Feige (see Feige, U. 2006. On sums of independent random variables with unbounded variances, and estimating the average degree in a graph. SIAM J. Comput. 35 964-984) can be obtained; also, these bounds 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.
AB - In this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount (relative to the expected value), by means of the second and the fourth (central) moments of the random variable. In this particular context, many classical inequalities yield only trivial bounds. We obtain tight upper bounds by studying the primal-dual moments-generating conic optimization problems. As an application, we demonstrate that given the new probability bounds, a substantial sharpening and simplification of a recent result and its analysis by Feige (see Feige, U. 2006. On sums of independent random variables with unbounded variances, and estimating the average degree in a graph. SIAM J. Comput. 35 964-984) can be obtained; also, these bounds 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.
KW - Duality
KW - Moments problem
KW - Probability of small deviation
KW - Sum of random variables
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U2 - 10.1287/moor.1090.0438
DO - 10.1287/moor.1090.0438
M3 - Article
AN - SCOPUS:77949389600
SN - 0364-765X
VL - 35
SP - 208
EP - 232
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 1
ER -