Some asymptotic results related to the law of iterated logarithm for Brownian motion |
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Authors: | Terence Chan |
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Institution: | (1) Department of Actuarial Mathematics and Statistics, Heriot-Watt University, EH14 4AS Edinburgh, UK |
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Abstract: | For 0< <1, let
. The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup
t![rarr](/content/354418833676653k/xxlarge8594.gif)
U
((t))=1–exp{–4( –1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU
(t), t>0. Also, when =1,U
(t) 0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form
as ![lambda](/content/354418833676653k/xxlarge955.gif) 0, whereD is a suitable discount function. These results also hold for symmetric random walks. |
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Keywords: | Law of iterated logarithm |
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