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 is convergent with (finite) value R, then the series

converges uniformly in s ∈ [0, 1] and

If a_n ≥ 0 for all n and R = ∞, then lim_{n → ∞} R(s) = ∞.

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

THEOREM A.2.– (Tauber’s Theorem) If

and there exists a finite limit

then the sum is convergent and

DEFINITION A.1.– A positive function L(t), t ≥ t₀ is called slowly varying at infinity if

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) → ...