Skorohod Embedding and Invariance Principles
Embedding of random variables; approximation of random walks; functional central limit theorem; laws of the iterated logarithm; arcsine laws; approximation of renewal processes; empirical distribution functions; embedding and approximation of martingales
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- The first functional limit theorems were obtained in (1931b, 1933a) by Kol-Mogorov, who considered special functionals of a random walk. Erdös and Kac (1946, 1947) conceived the idea of an invariance principle that would allow functional limit theorems to be extended from particular cases to a general setting. They also treated some special functionals of a random walk. The first general functional limit theorems were obtained by Donsker (1951–52) for random walks and empirical distribution functions, following an idea of Doob (1949). A general theory based on sophisticated compactness arguments was later developed by Prohorov (1956) and others.Google Scholar
- Skorohod’s (1965) embedding theorem provided a new and probabilistic approach to Donsker’s theorem. Extensions to the martingale context were obtained by many authors, beginning with Dubins (1968). Lemma 14.19 appears in Dvoretzky (1972). Donsker’s weak invariance principle was supplemented by a strong version due to Strassen (1964), which yields extensions of many a.s. limit theorems for Brownian motion to suitable random walks. In particular, his result yields a simple proof of the Hartman and Wintner (1941) law of the iterated logarithm, which had originally been deduced from some deep results of Kolmogorov (1929).Google Scholar
- Billingsley (1968) gives many interesting applications and extensions of Donsker’s theorem. For a wide range of applications of the martingale embedding theorem, see Hall and Heyde (1980) and Durrett (1995). Komlós et al. (1975–76) showed that the approximation rate in the Skorohod embedding can be improved by a more delicate “strong approximation.” For an exposition of their work and its numerous applications, see Csörgö and Révész (1981).Google Scholar