Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

Request Permissions   Purchase Content 
 

 

A simplified version of Spitzer's formula for semicontinuous and almost semicontinuous processes


Author: D. V. Gusak
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
Journal: Theor. Probability and Math. Statist. 85 (2012), 61-71
MSC (2000): Primary 60G50; Secondary 60K10
Published electronically: January 11, 2013
MathSciNet review: 2933703
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{\xi (t), \zeta (0)=0, t\geq 0\}$ be a process with stationary independent increments. We establish simplified versions of relations for spectral functions in Spitzer's formulas in terms of the exponential function whose index is determined by the corresponding root of Lundberg's equation for the case where $ \xi (t)$ is a semicontinuous or almost semicontinuous process.


References [Enhancements On Off] (What's this?)

  • 1. Frank Spitzer, Principles of random walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1964. MR 0171290
  • 2. Frank Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339. MR 0079851, 10.1090/S0002-9947-1956-0079851-X
  • 3. D. V. Gusak, Granichni zadachi dlya protsesiv z nezalezhnimi prirostami v teorii riziku, \cyr Pratsī Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni. Matematika ta ïï Zastosuvannya [Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications], vol. 65, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 2007 (Ukrainian, with English and Ukrainian summaries). MR 2382816
  • 4. Julian Keilson, The first passage time density for homogeneous skip-free walks on the continuum, Ann. Math. Statist. 34 (1963), 1003–1011. MR 0153060
  • 5. V. M. Zolotarev, The moment of first passage of a level and the behaviour at infinity of a class of processes with independent increments, Teor. Verojatnost. i Primenen. 9 (1964), 724–733 (Russian, with English summary). MR 0171315
  • 6. A. A. Borovkov, On the first passage time for a class of processes with independent increments, Teor. Verojatnost. i Primenen. 10 (1965), 360–364 (Russian, with English summary). MR 0182052
  • 7. Søren Asmussen, Ruin probabilities, Advanced Series on Statistical Science & Applied Probability, vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794582

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60G50, 60K10

Retrieve articles in all journals with MSC (2000): 60G50, 60K10


Additional Information

D. V. Gusak
Affiliation: Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 252601 Kyiv–4, Ukraine
Email: random@imath.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2013-00874-6
Keywords: Semicontinuous and almost semicontinuous processes with independent increments, Spitzer’s formula, Lundberg’s equation, Spitzer’s spectral functions
Received by editor(s): December 10, 2010
Published electronically: January 11, 2013
Article copyright: © Copyright 2013 American Mathematical Society