A recurrence theorem for square-integrable martingales

Gerold Alsmeyer

Studia Mathematica (1994)

  • Volume: 110, Issue: 3, page 221-234
  • ISSN: 0039-3223

Abstract

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Let ( M n ) n ≥ 0 be a zero-mean martingale with canonical filtration ( ℱ n ) n ≥ 0 and stochastically L 2 -bounded increments Y 1 , Y 2 , . . . , which means that P ( | Y n | > t | ℱ n - 1 ) ≤ 1 - H ( t ) a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let V 2 = ∑ n ≥ 1 E ( Y n 2 | ℱ n - 1 ) . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. P ( M n ∈ [ - c , c ] i . o . | V = ∞ ) = 1 for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions. As an application of our recurrence theorem, we obtain an extension of Blackwell’s renewal theorem to a fairly general class of processes with independent increments and linear positive drift function.

How to cite

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Alsmeyer, Gerold. "A recurrence theorem for square-integrable martingales." Studia Mathematica 110.3 (1994): 221-234. <http://eudml.org/doc/216110>.

@article{Alsmeyer1994,
abstract = {Let $(M_n)_\{n≥0\}$ be a zero-mean martingale with canonical filtration $(ℱ_n)_\{n≥0\}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,..., $ which means that $P(|Y_n| > t | ℱ_\{n-1\}) ≤ 1 - H(t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2 = ∑_\{n≥1\} E(Y_\{n\}^\{2\}|ℱ_\{n-1\})$. It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P(M_\{n\} ∈ [-c,c] i.o. | V = ∞) = 1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions. As an application of our recurrence theorem, we obtain an extension of Blackwell’s renewal theorem to a fairly general class of processes with independent increments and linear positive drift function.},
author = {Alsmeyer, Gerold},
journal = {Studia Mathematica},
keywords = {martingale; stochastic $L_2$-boundedness; recurrence; first passage time; Blackwell's renewal theorem; coupling; zero-mean martingale; recurrence theorem},
language = {eng},
number = {3},
pages = {221-234},
title = {A recurrence theorem for square-integrable martingales},
url = {http://eudml.org/doc/216110},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Alsmeyer, Gerold
TI - A recurrence theorem for square-integrable martingales
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 3
SP - 221
EP - 234
AB - Let $(M_n)_{n≥0}$ be a zero-mean martingale with canonical filtration $(ℱ_n)_{n≥0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,..., $ which means that $P(|Y_n| > t | ℱ_{n-1}) ≤ 1 - H(t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2 = ∑_{n≥1} E(Y_{n}^{2}|ℱ_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P(M_{n} ∈ [-c,c] i.o. | V = ∞) = 1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions. As an application of our recurrence theorem, we obtain an extension of Blackwell’s renewal theorem to a fairly general class of processes with independent increments and linear positive drift function.
LA - eng
KW - martingale; stochastic $L_2$-boundedness; recurrence; first passage time; Blackwell's renewal theorem; coupling; zero-mean martingale; recurrence theorem
UR - http://eudml.org/doc/216110
ER -

References

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  1. [1] G. Alsmeyer, On the moments of certain first passage times for linear growth processes, Stochastic Process. Appl. 25 (1987), 109-136. Zbl0626.60037
  2. [2] G. Alsmeyer, Random walks with stochastically bounded increments: Renewal theory, submitted, 1991. 
  3. [3] G. Alsmeyer, Random walks with stochastically bounded increments: Renewal theory via Fourier analysis, submitted, 1991. 
  4. [4] R. Durrett, H. Kesten and G. Lawler, Making money from fair games, in: Random Walks, Brownian Motion and Interacting Particle Systems, R. Durrett and H. Kesten (eds.), Birkhäuser, Boston, 1991, 255-267. Zbl0746.60044
  5. [5] R. Gundy and D. Siegmund, On a stopping rule and the central limit theorem, Ann. Math. Statist. 38 (1967), 1915-1917. Zbl0155.25401
  6. [6] P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980. Zbl0462.60045
  7. [7] J. H. B. Kemperman, The oscillating random walk, Stochastic Process. Appl. 2 (1974), 1-29. Zbl0326.60081
  8. [8] H. Kesten and G. F. Lawler, A necessary condition for making money from fair games, Ann. Probab. 20 (1992), 855-882. Zbl0758.60067
  9. [9] S. P. Lalley, A first-passage problem for a two-dimensional controlled random walk, J. Appl. Probab. 23 (1986), 670-678. Zbl0612.60059
  10. [10] J. Lamperti, Criteria for the recurrence or transience of stochastic processes I, J. Math. Anal. Appl. 1 (1960), 314-330. Zbl0099.12901
  11. [11] B. A. Rogozin and S. G. Foss, Recurrency of an oscillating random walk, Theor. Probab. Appl. 23 (1978), 155-162. Zbl0423.60059

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