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

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  • de la Peña, Victor H.
  • Govindarajulu, Z.

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).

Suggested Citation

  • de la Peña, Victor H. & Govindarajulu, Z., 1992. "A note on second moment of a randomly stopped sum of independent variables," Statistics & Probability Letters, Elsevier, vol. 14(4), pages 275-281, July.
    • Handle: RePEc:eee:stapro:v:14:y:1992:i:4:p:275-281
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    Cited by:

    1. G. Celik & S. C. Borst & P. A. Whiting & E. Modiano, 2016. "Dynamic scheduling with reconfiguration delays," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 87-129, June.

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