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:

Image

where a(t) → c ∈ (0, ∞) and ε(t) → 0 as t.

A.1. Examples of slowly varying functions

Image

and

Image

The fact that g(t) is a slowly varying function which follows from Proposition A.1 and the equality:

Image

where

Image

as t. Note that:

Image

Thus, slowly varying functions may oscillate in a rather exotic manner. □

THEOREM A.3.– (Tauberian Theorem) We assume an ≥ 0 and let the series R(s) = image converge for s ∈ [0, 1). Then, the following statements are equivalent for ρ ∈ [0, ∞) :

and

Image

If an is monotonic and ρ ∈ (0, ∞), then [A.1] is equivalent to

Image

Moreover, if H is a measure on [0, ∞) such that for H(t) := H([0, t])

image

then

Image

Let image be the distribution of a non-negative random variable η.

DEFINITION A.2.– The function

Image

where the symbol ∗ means the convolution, is called the renewal function corresponding to G.

We assume that:

THEOREM A.4.– If condition [A.2] holds, then, as t,

Image

PROOF.– We have:

Image

Thus,

Image

given by the Tauberian theorem with ρ = 1 that

Image

This is our claim. □

THEOREM A.5. (see Theorem 1.4.6 in [BOR 08]) Let

Image

be a function such that hn ≥ 0 and

Image

for some α > 1 and a function l(n) slowly varying at infinity. If image(w) is an analytical function in a domain containing the circle

Image

then

Image

and

Image

as n.

LEMMA A.1.– Let (βn, n ≥ 0) be a regularly varying sequence which fulfills the inequality image. If image, then image with image.

LEMMA A.2.– (The strong law of large numbers, see Theorem 13, Chapter IX.3 in [PET 75]) Let ζ1, ζ2, … be a sequence of independent and identically distributed random variables. If image for some p ∈ (0, 1), then for any ε > 0, as n,

Image
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