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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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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 (S
1, S
2, …, 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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Geometrically stopped random walk. ADV APPL PROBAB 2016. [DOI: 10.1017/s0001867800030020] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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Greenwood P, Omey E, Teugels JL. Harmonic renewal measures and bivariate domains of attraction in fluctuation theory. Probab Theory Relat Fields 1982. [DOI: 10.1007/bf00531622] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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