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1994 | 110 | 3 | 221-234
Tytuł artykułu

A recurrence theorem for square-integrable martingales

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Abstrakty
EN
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.
Twórcy
  • Mathematisches Seminar, Universität Kiel, Ludewig-Meyn-Strasse 4, D-2300 Kiel, 1, Germany
Bibliografia
  • [1] G. Alsmeyer, On the moments of certain first passage times for linear growth processes, Stochastic Process. Appl. 25 (1987), 109-136.
  • [2] G. Alsmeyer, Random walks with stochastically bounded increments: Renewal theory, submitted, 1991.
  • [3] G. Alsmeyer, Random walks with stochastically bounded increments: Renewal theory via Fourier analysis, submitted, 1991.
  • [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.
  • [5] R. Gundy and D. Siegmund, On a stopping rule and the central limit theorem, Ann. Math. Statist. 38 (1967), 1915-1917.
  • [6] P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
  • [7] J. H. B. Kemperman, The oscillating random walk, Stochastic Process. Appl. 2 (1974), 1-29.
  • [8] H. Kesten and G. F. Lawler, A necessary condition for making money from fair games, Ann. Probab. 20 (1992), 855-882.
  • [9] S. P. Lalley, A first-passage problem for a two-dimensional controlled random walk, J. Appl. Probab. 23 (1986), 670-678.
  • [10] J. Lamperti, Criteria for the recurrence or transience of stochastic processes I, J. Math. Anal. Appl. 1 (1960), 314-330.
  • [11] B. A. Rogozin and S. G. Foss, Recurrency of an oscillating random walk, Theor. Probab. Appl. 23 (1978), 155-162.
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