A note on second moment of a randomly stopped sum of independent variables

Victor H. de la Peña and Z. Govindarajulu

Statistics & Probability Letters, 1992, vol. 14, issue 4, 275-281

Abstract: Let X1, X2,... be a sequence of independent random variables having finite second moments. Let T be a stopping time defined on the X's. If is an integer-valued random variable independent of the X's but having the same distribution as T, then it is shown that 0 [less-than-or-equals, slant] ES2T [less-than-or-equals, slant] 2 ES2 where Sn = X1 + ... + Xn. Further, via examples it is shown that the above lower and upper bounds are sharp. Some different bounds when the variables are nonnegative are also given. For a sequence of independent random variables, upper and lower bounds for the expectation of a non-negative function of the randomly stopped sequence are obtained (see Result 4).

Date: 1992
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