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A note on a condition satisfied by certain random walks
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- Journal:
- Journal of Applied Probability / Volume 14 / Issue 4 / December 1977
- Published online by Cambridge University Press:
- 14 July 2016, pp. 843-849
- Print publication:
- December 1977
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- Article
- Export citation
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The problem considered is to elucidate under what circumstances the condition
holds, where 
and Xi are independent and have common distribution function F. The main result is that if F has zero mean, 
and 
(*) holds with 
F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases 
are also treated. (The case 
cannot arise in these circumstances.) A partial result is also given for the case when 
and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn, the number of positive terms in (S1, S2, …, Sn). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.
and 
and 
(*) holds with 
are also treated. (The case 
cannot arise in these circumstances.) A partial result is also given for the case when 
and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for 