The speed of convergence of a martingale |
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Authors: | Harry Kesten |
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Institution: | (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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Keywords: | |
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