×

Unusually large values for spectrally positive stable and related processes. (English) Zbl 0942.60029

Author’s summary: Two classes of processes are considered. One is a class of spectrally positive infinitely divisible processes which includes all such stable processes. The other is a class of processes constructed from the sequence of partial sums of independent identically distributed positive random variables. A condition analogous to regular variation of the tails is imposed. Then a large deviation principle and a Strassen-type law of the iterated logarithm are presented. These theorems focus on unusually large values of the processes. They are expressed in terms of Skorokhod’s \(M_1\) topology.

MSC:

60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[2] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York. · Zbl 0592.60049
[3] It o, K. (1942). On stochastic processes I. Japan J. Math. 18 261-301. · Zbl 0060.28908
[4] Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Probab. 15 610-627. · Zbl 0624.60045
[5] O’Brien, G. L. (1996). Sequences of capacities, with connections to large deviation theory. J. Theoret. Probab. 9 19-35. · Zbl 0847.60061
[6] O’Brien, G. L. and Vervaat, W. (1996). Large deviation principles and loglog laws for observation processes. Statist. Neerlandica 50 242-260. (Memorial issue for W. Vervaat.) · Zbl 0858.60030
[7] Pakshirajan, R. P. and Vasudeva, R. (1981). A functional law of the iterated logarithm for a class of subordinators. Ann. Probab. 9 1012-1018. · Zbl 0477.60035
[8] Schilder, M. (1966). Some asymptotics formulae for Wiener integrals. Trans. Amer. Math. Soc. 125 63-85. · Zbl 0156.37602
[9] Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theoret. Probab. Appl. 1 261- 290. · Zbl 0074.33802
[10] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm.Wahrsch. Verw. Gebiete 3 211-226. · Zbl 0132.12903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.