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Convolution equivalence and distributions of random sums. (English) Zbl 1146.60014
Summary: A serious gap in the proof of A. G. Pakes’s paper [J. Appl. Probab. 41, No. 2, 407–424 (2004; Zbl 1051.60019)] on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev’s theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Lévy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin’s theorem is proved.

##### MSC:
 60E07 Infinitely divisible distributions; stable distributions 60G50 Sums of independent random variables; random walks 60G51 Processes with independent increments; Lévy processes
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