Spitzer's condition for asymptotically symmetric random walk
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If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that lim x→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.
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1973 ◽
Vol 16
(2)
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pp. 173-177
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1998 ◽
Vol 12
(3)
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pp. 373-386
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2017 ◽
Vol 69
(1)
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pp. 110-128
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1977 ◽
Vol 14
(04)
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pp. 843-849
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2004 ◽
Vol 41
(03)
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pp. 623-638
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1982 ◽
Vol 32
(3)
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pp. 412-422
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