On divergence and convergence of sums of nonnegative random variables

J.C. Smit and W. Vervaat

Statistica Neerlandica, 1983, vol. 37, issue 3, 143-147

Abstract: Abstract If X1, X2,… are exponentially distributed random variables thenσ∞k= 1 Xk=∞ with probability 1 iff σ∞k= 1 EXk=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI‐MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.

Date: 1983
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