Boundary functionals for the superposition of a random walk and a sequence of independent random variables

Authors:
I. I. Ezhov and V. F. Kadankov

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **75** (2006).

Journal:
Theor. Probability and Math. Statist. **75** (2007), 9-22

MSC (2000):
Primary 60J05, 60J10; Secondary 60J45

Published electronically:
January 23, 2008

MathSciNet review:
2321177

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For the superposition of a random walk and a sequence of independent random variables, we obtain the moment generating functions of the joint distribution of the first passage time and overshoot over a level, and those of the joint distribution of the first exit time from an interval and the value of the superposition at the exit time.

**1.**E. A. Pečerskiĭ and B. A. Rogozin,*The combined distributions of the random variables connected with the fluctuations of a process with independent increments*, Teor. Verojatnost. i Primenen.**14**(1969), 431–444 (Russian, with English summary). MR**0260005****2.**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****3.**A. A. Borovkov,*Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya*, Izdat. “Nauka”, Moscow, 1972 (Russian). MR**0315800****4.**V. F. Kadankov and T. V. Kadankova,*On the distribution of the first exit time from an interval and the value of the overjump across a boundary for processes with independent increments and random walks*, Ukraïn. Mat. Zh.**57**(2005), no. 10, 1359–1384 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**57**(2005), no. 10, 1590–1620. MR**2219768**, 10.1007/s11253-006-0016-6**5.**I. G. \cyr{P}etrovskiĭ,*Lektsii po teorii integralnykh uravnenii*, Izdat. “Nauka”, Moscow, 1965 (Russian). Third edition, revised. MR**0352904**

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Additional Information

**I. I. Ezhov**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, 01601, Kyiv, Ukraine

**V. F. Kadankov**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, 01601, Kyiv, Ukraine

Email:
kadankov@voliacable.com

DOI:
https://doi.org/10.1090/S0094-9000-07-00710-7

Keywords:
Boundary functionals,
first exit time from an interval for a random walk,
the superposition of a random walk and a sequence of independent random variables

Received by editor(s):
April 25, 2005

Published electronically:
January 23, 2008

Article copyright:
© Copyright 2007
American Mathematical Society