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Discrete Time Branching Processes in Random Environment
book

Discrete Time Branching Processes in Random Environment

by Götz Kersting, Vladimir Vatutin
November 2017
Intermediate to advanced content levelIntermediate to advanced
306 pages
6h 24m
English
Wiley-ISTE
Content preview from Discrete Time Branching Processes in Random Environment

Appendix

For the convenience of reference, we list a number of classical results in this section.

THEOREM A.1.– (Abel’s Theorem) If the series image is convergent with (finite) value R, then the series

Image

converges uniformly in s ∈ [0, 1] and

Image

If an ≥ 0 for all n and R = ∞, then limn → ∞ R(s) = ∞.

With the same notation as in the previous theorem, the following statement is valid.

THEOREM A.2.– (Tauber’s Theorem) If

Image

and there exists a finite limit

Image

then the sum image is convergent and

Image

DEFINITION A.1.– A positive function L(t), tt0 is called slowly varying at infinity if

Image

PROPOSITION A.1.– A function L(t) is slowly varying at infinity if and only if it may be represented in the form:

where a(t) → ...

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Publisher Resources

ISBN: 9781786302526Purchase book