Random Walks and Renewal Theory

• Olav Kallenberg
Chapter
Part of the Probability and Its Applications book series (PIA)

Abstract

Recurrence and transience; dependence on dimension; general recurrence criteria; symmetry and duality; Wiener-Hopf factorization; ladder time and height distribution; stationary renewal process; renewal theorem

References

1. Random walks originally arose in a wide range of applications, such as gambling, queuing, storage, and insurance; their history can be traced back to the origins of probability. The approximation of diffusion processes by random walks dates back to Bachelier (1900, 1901). A further application was to potential theory, where in the 1920s a method of discrete approximation was devised, admitting a probabilistic interpretation in terms of a simple symmetric random walk. Finally, random walks played an important role in the sequential analysis developed by Wald (1947).Google Scholar
2. The modern theory began with Polya’s (1921) discovery that a simple symmetric random walk on ℤd is recurrent for d ℤ 2 and transient otherwise. His result was later extended to Brownian motion by Lévy (1940) and Kakutani (1944a). The general recurrence criterion in Theorem 9.4 was derived by Chung and Fuchs (1951), and the probabilistic approach to Theorem 9.2 was found by Chung and Ornstein (1962). The first condition in Corollary 9.7 is, in fact, even necessary for recurrence, as was noted independently by Ornstein (1969) and C.J. Stone (1969).Google Scholar
3. The reflection principle was first used by Andre (1887) in his discussion of the ballot problem. The systematic study of fluctuation and absorption problems for random walks began with the work of Pollaczek (1930). Ladder times and heights, first introduced by Blackwell, were explored in an influential paper by Feller (1949). The factorizations in Theorem 9.15 were originally derived by the Wiener-Hopf technique, which had been developed by Paley and Wiener (1934) as a general tool in Fourier analysis.Google Scholar
4. Theorem 9.16 is due for u = 0 to Sparre-Andersen (1953–54) and in general to Baxter (1961). The former author used complicated combinatorial methods, which were later simplified by Feller and others.Google Scholar
5. Though renewals in Markov chains are implicit already in some early work of Kolmogorov and Levy, the general renewal process was apparently first introduced by Palm (1943). The first renewal theorem was obtained by Erdös et al. (1949) for random walks on Z+. In that case, however, Chung noted that the result is an easy consequence of Kol-mogorov’s (1936a-b) ergodic theorem for Markov chains on a countable state space. Blackwell (1948,1953) extended the result to random walks on ℝ+. The ultimate version for transient random walks on R is due to Feller and Orey (1961). The first coupling proof of Blackwell’s theorem was given by Lindvall (1977). Our proof is a modification of an argument by Athreya et al. (1978), which originally did not cover all cases. The method seems to require the existence of a possibly infinite mean. An analytic approach to the general case appears in Feller (1971).Google Scholar
6. Elementary introductions to random walks are given by many authors, including Chung (1974), Feller (1968, 1971), and Loève (1977). A detailed exposition of random walks on d is given by Spitzer (1976).Google Scholar