The speed of convergence of a martingale |
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Authors: | Harry Kesten |
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Affiliation: | (1) Department of Mathematics, Cornell University, 14853 Ithaca, New York, USA |
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Abstract: | LetX n, n≧0, be a martingale with respect to the σ-fieldsF n and letB n 2 =Σ1≧n E{(X 1−X 1−1)2|F 1−1} It is known that ifB 1 2 <∞ on some set Ω0 thenX ∞=limX n exists and is finite a.e. on Ω0 We show that under suitable conditions there exists a constant ν<∞ for which lim supB n −1 {log logB n 2 }−1/2|X ∞−X n−1 | ≦ √2(η+1). If “the fluctuations ofB n are small” (in the sense of the Corollary) then ν=0 and the usual upper bound of a law of the iterated logrithm results. This upper bound is not necessarily achieved, though. Research supported in part by the NSF under Grant No. MCS 72-04534A04. |
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