Results in Mathematics

, Volume 19, Issue 1–2, pp 22–45 | Cite as

Random Walks With Stochastically Bounded Increments: Foundations and Characterization Results

  • Gerold Alsmeyer


We consider random walks S, adapted to a filtration \(\cal F_N\), whose conditional increment distribution functions are bounded from above and/or below by an integrable distribution function. A further stability condition on the conditional increment means is also introduced. Such random walks share a number of properties with those having i.i.d. increments, in particular a uniform law of large numbers. In this paper, which is accompanied by a second one on renewal theory, we derive their basic properties and give equivalent characterizations in terms of certain drift constants which are introduced before and of great importance for a renewal theoretic analysis.


Random Walk Canonical Representation Entrance Time Renewal Theory Maximal Minorant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alsmeyer, G.(1987). On the moments of certain first passage times for linear growth processes. Stoch. Proc. Appl. 25 109–136.Google Scholar
  2. [2]
    Alsmeyer, G.(1990). Random walks with stochastically bounded increments: Renewal theory. To appear in Ann. Probab. Google Scholar
  3. [3]
    Gänssler, P. and Stute, W.(1977). Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar
  4. [4]
    Klass, M.J.(1973). Properties of optimal extended-valued stopping rules for Sn/n. Ann. Probab. 1 719–757.Google Scholar
  5. [5]
    Neveu, J.(1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
  6. [6]
    Pitman, J. and Speed, T.(1973). A note on random times. Stoch. Proc. Appl. 1 369–374.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Smith, W.(1961). On some general renewal theorems for nonidentically distributed random variables. Proc. 4th Berkeley Symp. on Math. Stat. and Prob., Vol. 2 467–514.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1991

Authors and Affiliations

  • Gerold Alsmeyer
    • 1
  1. 1.Mathematisches SeminarUniversität KielKiel 1

Personalised recommendations