On pairs of sums of random variables with pairwise equal distributions

U. Krengel

Statistics & Probability Letters, 1996, vol. 27, issue 3, 217-219

Abstract: Let X1, X2, Y1, Y2 be four real-valued random variables such that X1, X2 have a common distribution function F, and Y1 and Y2 have a common distribution function G. Furstenberg (1967) asked: If X1+y1[greater-or-equal, slanted]X2+y2 holds, does it follow that equality holds with probability 1? Obviously, the answer is positive if the expectations of X1 and Y1 exist. We show that the existence of the expectation of X1 is sufficient. We also show by example (announced by Krengel (1970)) that the answer is negative if both X1 and Y1 are allowed to have infinite expectation, even if all random variables are nonnegative.

Date: 1996
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