A non-Markovian birth process with logarithmic growth
Keyword(s):
I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 < λ < 1, is an example of such a process.
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1983 ◽
Vol 94
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pp. 251-263
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1995 ◽
Vol 118
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pp. 527-542
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1968 ◽
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pp. 485-488
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1992 ◽
Vol 112
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pp. 613-629
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1963 ◽
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pp. 411-416
1980 ◽
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pp. 179-187
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1971 ◽
Vol 8
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pp. 52-59
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