A note on a condition satisfied by certain random walks

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Abstract
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.

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