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 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
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) → c ∈ (0, ∞) and ε(t) → 0 as t → ∞.
A.1. Examples of slowly varying functions
and
The fact that g(t) is a slowly varying function which follows from Proposition A.1 and the equality:
where
as t → ∞. Note that:
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) = 
converge for s ∈ [0, 1). Then, the following statements are equivalent for ρ ∈ [0, ∞) :
and
If an is monotonic and ρ ∈ (0, ∞), then [A.1] is equivalent to
Moreover, if H is a measure on [0, ∞) such that for H(t) := H([0, t])
then
Let 
be the distribution of a non-negative random variable η.
DEFINITION A.2.– The function
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 → ∞,
PROOF.– We have:
Thus,
given by the Tauberian theorem with ρ = 1 that
This is our claim. □
THEOREM A.5. (see Theorem 1.4.6 in [BOR 08]) Let
be a function such that hn ≥ 0 and
for some α > 1 and a function l(n) slowly varying at infinity. If 
(w) is an analytical function in a domain containing the circle
then
and
as n → ∞.
LEMMA A.1.– Let (βn, n ≥ 0) be a regularly varying sequence which fulfills the inequality 
. If 
, then 
with 
.
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 
for some p ∈ (0, 1), then for any ε > 0, as n → ∞,
