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An improvement of Watson’s theorem on Borel summability
Journal of Mathematical Physics, 1980Watson’s theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector ‖argz‖<π/2+ε, ‖z‖<R, with ε≳0; in fact, only the circular region Re(1/z) ≳1/R is required. In particular, one can take ε=0.
Alan D Sokal, Sokal Alan D
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On a Theorem of Quine and Seneta for the Galton‐Watson Process With Immigration
The Australian Journal of Statistics, 1971SummaryRecently a limit theorem has been obtained for the limiting‐stationary distribution of a process in which individuals reproduce as in a subcritical Galton‐Watson process and are subject to an independent immigration component at each generation. This paper provides a different proof of this theorem, and under slightly weaker conditions.
A G Pakes
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Limit Theorems for a Galton–Watson Process with Immigration in Varying Environments
Bulletin of the Malaysian Mathematical Sciences Society, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Limit theorem for a supercritical Galton-Watson process
Mathematical Notes, 1975Letμ n, n = 0, 1, ..., be a Galton-Watson process, and τx + 1 the instant of first crossing of the level x by the process. A limit theorem is proved for the joint distribution of the random variables $$\tau _x ,x - \mu _{\tau _x } ,\mu _{\tau _x + 1} - x(x \to \infty )$$
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A New Proof of Extended Watson Summation Theorem
In the theory of generalized hypergeometric series, classical Watson summation theorem play an important role. In 2010, Kim et al. have given two extensions of the classical Watson summation theorem. In this note, we aim to provide a new proof for one of
Sungtae Jun, Adem Kılıc
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