On the sum of nonnegative independent random variables with unbounded variance

We prove the following inequality: for every positive integer n and every collection X 1 ; : : : ; X n of nonnegative independent random variables that each has expectation 1, the probability that their sum exceeds n+1 is at most < 1. Our proof produces a value of = 12=13 ' 0:923, but we conjecture that the inequality also holds with = 1 ? 1=e ' 0:632. 1 A new inequality For a random variable X, its typical value may be very diierent from its mean. In particular, the probability that X exceeds its mean may be arbitrarily close to 1. In some special case (e.g., when X is symmetric around its mean), the probability that X exceeds its mean is at most 1=2. The purpose of this manuscript is to investigate the probability that X exceeds its mean when X is the sum of n independent random variables. We show that under very general conditions, this probability is bounded away from 1, provided that we give ourselves a little slackness in exceeding the mean. Speciically, we prove the following inequality concerning the sum of independent nonnegative random variables. (1) 1 The term =(1+) in Theorem 1 is best possible, as one can take X 1 = 1 + with probability 1=(1 +) and 0 otherwise, and all of the other X i as the constant 1. This gives i = 1 for every i. For this case PrX < +] = PrX 1 = 0] = =(1+). When is above some small constant (1=12 in the current statement of the theorem) the bounds in Theorem 1 are no longer tight. Moreover, for even larger (e.g., = 1), it is no longer true that PrX +] =(1+). One can take for every i, X i = n+ with probability 1=(n +) and 0 otherwise. This gives i = 1 for every i, implying = n. For this case PrX < n + ] = (1 ? 1=(n +)) n , which is roughly 1=e for large n. It is our conjecture that for every value of and n, one of the two examples above is the worst case for PrX < +]. The conjecture, if true, would allow us to replace the constant 1=13 by 1=e in Theorem 1. In may be instructive to consider how some standard probabilistic tools relate to Theorem 1. Consider rst the case …