Regularly distributed randomly stopped sum, minimum, and maximum ∗

. Let { ξ 1 , ξ 2 , . . . } be a sequence of independent real-valued, possibly nonidentically distributed, random variables, and let η be a nonnegative, nondegenerate at 0 , and integer-valued random variable, which is independent of { ξ 1 , ξ 2 , . . . } . We consider conditions for { ξ 1 , ξ 2 , . . . } and η under which the distributions of the randomly stopped minimum, maximum, and sum are regularly varying.

Before discussing the properties of the d.f.s F ξ (η) , F ξ (η) , and F Sη , we recall the definition of a regularly varying distribution function.
A d.f. F is regularly varying (F ∈ R α ) for some index α 0 if lim x→∞ F (xy) F (x) = y −α for any y > 0. A r.v. X is regularly varying with index α 0 if its distribution F X belongs to R α .
An important property following from the definition of R α is that the tail function of an arbitrary regularly varying d.f. can be represented in the form where L is a slowly varying function, that is, for any y > 0.
In this paper, we consider a sequence {ξ 1 , ξ 2 , . . .} of possibly nonidentically distributed r.v.s. We suppose that some of the d.f.s of these r.v.s belong either to the class R α with some α 0 or to the classes R = α 0 R α , R + = α>0 R α . We find conditions under which the d.f.s F ξ (η) , F ξ (η) , and F Sη are regularly varying.
The class of regularly varying functions was introduced by Karamata [14] in the context of real analysis. The notion of regular variation was introduced in probability theory http://www.journals.vu.lt/nonlinear-analysis by Feller [11] when considering limit theorems for sums of i.i.d. r.v.s. Many analytical results on regularly varying functions can be found in the monograph by Bingham et al. [1]. Some applications of regularly varying distributions to finance and insurance are presented by Embrechts et al. [9]. We further present a few results on randomly stopped structures related to regularly varying distribution functions.
The following result on sufficient and necessary conditions for the closure under random maximum of regularly varying r.v.s was obtained in [13] (see Lemma 5.1(i)).
Motivated by the presented statements and results obtained in [2, 4, 5, 7, 15, 18-21, 24, 26], we continue to consider conditions under which the d.f.s F ξ (η) , F ξ (η) , and F Sη belong to either the class R α with some α 0 or the class R. As we mentioned before, we deal with the case where the sequence {ξ 1 , ξ 2 , . . .} consists of independent but possibly nonidentically distributed r.v.s.
The rest of the paper is organized as follows. In Section 2, we present our main results. Section 3 consists of some auxiliary lemmas. The proofs of the main results are given in Section 4. Finally, in Section 5, we present two examples to expose the usefulness of our results.

Main results
In this section, we present the main results of this paper. In all the statements, we suppose that the sequence {ξ 1 , ξ 2 , . . . } and the c.r.v. η are independent. Our first theorem describes properties of randomly stopped minima. Theorem 4. Let {ξ 1 , ξ 2 , . . .} be a sequence of real-valued r.v.s. Then F ξ k ∈ R for all k ∈ N if and only if F ξ (η) ∈ R for every c.r.v. η.
The second theorem below describes sufficient conditions for the regularity of randomly stopped maxima and sums when the c.r.v. η has a finite support. Theorem 5. Let ξ 1 , . . . , ξ m be independent real-valued r.v.s, and let η be a counting r.v. independent of {ξ 1 , . . . , ξ m } such that P(η m) = 1. Then the d.f.s F Sη and F ξ (η) belong to the class R α , α 0, if the following two conditions are satisfied: Furthermore, under conditions (i)-(ii), the following tail equivalences hold: where I n = {k = 1, . . . , n: F ξ k ∈ R α }, and L k are slowly varying functions from the The following theorem describes properties of randomly stopped sums and maxima when the c.r.v. has a finite support. Here we provide both necessary and sufficient conditions for {ξ 1 , ξ 2 , . . . }, but the initial conditions for the collection of the primary r.v.s are more restrictive than in the previous theorem.
Then the following statements are equivalent: (ii) F Sη ∈ R + for any c.r.v. η with finite support; (iii) F ξ (η) ∈ R + for any c.r.v. η with finite support.
In the following theorem, we give sufficient conditions under which the randomly stopped sum is regularly varying in the case of a general c.r.v. η. Theorem 7. Let ξ 1 , ξ 2 , . . . independent real-valued r.v.s, and let η be a counting r.v. independent of {ξ 1 , ξ 2 , . . .}. Assume the following conditions are satisfied: Then the following tail equivalence holds: and hence F Sη ∈ R α . http://www.journals.vu.lt/nonlinear-analysis The following result on sufficient and necessary conditions for the closure under random maximum of regularly varying r.v.s is a direct generalization of Theorem 3.

Auxiliary lemmas
In this section, we give several auxiliary lemmas. Some of these lemmas are originally stated for wider heavy-tailed distribution classes, which include the class R as a subclass.
Here we restate these lemmas for regularly varying d.f.s. The first lemma follows from Theorem 3.1 of [3] (see also Theorem 2.1 from [25]).
The next lemma is Theorem 4.1 from [23]. This lemma provides necessary and sufficient conditions for the max-sum equivalence of regularly varying distributions.
In this case, the asymptotic relation (2) holds.
The next lemma follows from Theorems 3.10, 3.11, and 4.1 by Shimura [23]. It describes the decomposition property of regularly varying distributions.
Lemma 3. Let X be a real-valued r.v., and suppose that F X ∈ R + . Furthermore, suppose that X can be decomposed into independent r.v.s X 1 and X 2 , that is, The following statement was proved in Proposition 1 of [8] and later was generalized to a broader distribution class in Corollary 3.19 of [12]. Lemma 4. Let {X 1 , . . . , X n } be a collection of independent real-valued r.v.s. Assume that F X k (x)/F (x) → b k as x → ∞ for some regularly varying d.f. F and some constants b i 0, i ∈ {1, . . . n}. Then In the next lemma, we show in which cases the d.f. F Σn of the sum Σ n = X 1 + · · · + X n and the d.f. F X (n) of the maximum X (n) = max{X 1 , . . . , X n } belong to the class R α .
Lemma 5. Let X 1 , . . . , X n be independent real-valued r.v.s. Then the d.f.s F Σn and F X (n) belong to the class R α , α 0, if the following conditions are satisfied: Furthermore, under these conditions, where L k are slowly varying functions from representations F X k (x) = x −α L k (x), and I n = {k = 1, . . . , n: F X k ∈ R α }.
Proof. We first consider the sum Σ n . For n = 2, the statement is well known (see, e.g., p. 278 in [11], Lemma 1.3.4 in [17], Proposition 4.2.5 in [22] or the case n = 2 of Corollary 3.19 of [12]). We use induction. Suppose the statement of the lemma holds for n = K. This means that F Σ K ∈ R α and, due to Lemma 2, According to the conditions of the lemma, either F X K+1 ∈ R α or F X K+1 (x) = o(F Σ K (x)). Since Σ K+1 = Σ K + X K+1 , in both cases, we obtain that F Σ K+1 ∈ R α and by Proposition 4.2.5 from [22] and Proposition 1.3.6 from [1] on the properties of slowly varying functions. According to the induction principle, the statement of the lemma holds for all sums Σ n .
The statement of the lemma for X (n) follows immediately from the following asymptotic relations: http://www.journals.vu.lt/nonlinear-analysis for each n ∈ N by the classical Bonferoni inequalities and properties of slowly varying functions. The lemma is proved.
The following statement follows from Lemma 3.2 of [6]. Lemma 6. Let X 1 , . . . , X n be independent real-valued r.v.s, and let F Xν ∈ R α for some ν 1 and α 0. Suppose, in addition, that lim sup Then, for any p > α, there exists a positive constant c = c(p) such that for all n ν and x 0.
In fact, Lemma 3.2 in [6] is proved for nonnegative r.v.s, but the statement remains valid for real-valued r.v.s. To see this, it suffices to observe that P(X 1 + · · · + X n > x) P(X + 1 · · · + X + n > x) and P(X k > x) = P(X + k > x) for n ∈ N, k ∈ {1, . . . , n}, x 0, where a + denotes the positive part of a.

Proofs of main results
In this section, we give detailed proofs of our main results.
Then for any x > 0, we have and Therefore we obtain Necessity. If F ξ k ∈ R for all k ∈ N, then F ξ1 ∈ R α1 , F ξ2 ∈ R α2 , . . . for some nonnegative parameters α 1 , α 2 , . . . . This means that, for each k ∈ N, F ξ k (x) = x −α k L k (x) with a slowly varying function L k . Hence, for a finite nonrandom κ, by the closure properties of slowly varying functions (see, e.g., Proposition 1.3.6 in Bingham et al. [1]) because for x > 0. Thus, it follows from (3) and (4) that Sufficiency. If F ξ (η) ∈ R for an arbitrary c.r.v. η, then from (3) it follows that F ξ (n) ∈ R for any fixed n ∈ N. In addition, for all x > 0, we have that F ξ1 (x) = F ξ (1) (x) and Therefore, by the closure properties of slowly varying functions (see, e.g., Proposition 1.3.6 in Bingham et al. [1]), we obtain that F ξ k ∈ R for each k ∈ N. Theorem 4 is proved.
Proof of Theorem 5. To verify that F Sη ∈ R α , it suffices to prove that for some slowly varying function L.
For all x > 0, we have By Lemma 5 we conclude that for each n ∈ {1, . . . , m}, where L k are slowly varying functions. Asymptotic relation (1) now immediately follows.
http://www.journals.vu.lt/nonlinear-analysis By the closure properties of slowly varying functions (see Proposition 1.3.6 in Bingham et al. [1]) we conclude that asymptotic relation (5) holds with slowly varying function The proof of the theorem for the d.f. F ξ (η) is identical to that for F Sη , and hence we omit it. The theorem is proved.
Suppose now assumption (iii) holds, that is, F ξ (η) ∈ R + for any c.r.v. η with finite support. From this assumption it follows that F ξ (n) ∈ R αn for each n ∈ N with some index α n > 0. Applying the classical Bonferroni inequality, we obtain Therefore the d.f. max{0, 1 − n k=1 F k } belongs to the class R αn as well. Lemma 2 and the last asymptotic relation imply that F Sn ∈ R αn and for n ∈ N. Let us consider a c.r.v. η with finite support {0, 1, . . . , m}, m 1. In such a case, by the asymptotic relation (6) we have Consequently, F Sη ∈ R + for c.r.v. η. The implication (iii) ⇒ (ii) is proved.
Finally, we give a proof of the implication (ii) ⇒ (i). Since by assumption (ii) F Sη ∈ R + for every c.r.v. η with finite support, it follows that for each n ∈ N. In particular, F ξ1 ∈ R + and F ξ1+ξ2 ∈ R + . Lemma 3 implies that F ξ1 ∈ R α and F ξ2 ∈ R α for some α > 0 because F ξ1 (x) F ξ2 (x) by the conditions of the theorem. Let us continue by induction. Suppose that F ξ1 ∈ R α , F ξ2 ∈ R α , . . . , F ξ K ∈ R α with K 2. Lemma 1 implies that F S K ∈ R α ⊂ R + and The distribution function F S K+1 ∈ R + because of relation (7). In addition, the conditions of the theorem imply that F ξ k (x) F ξ1 (x) F ξ K+1 (x) for each k ∈ {1, . . . , K}. This, together with asymptotic relation (8), implies that F S K (x) F ξ K+1 (x). Using Lemma 3, we obtain that F ξ K+1 ∈ R α ⊂ R + . Now statement (i) of Theorem 6 follows by the induction principle. This completes the proof.
Proof of Theorem 7. As in Theorem 5, it suffices to prove the tail equivalence formula (5) with some slowly varying function L. For any K ∈ N and all x > 0, define the function In addition, for all x > 0, define We begin with the existence of this limit. First, for each fixed x, the sequence L * K (x) is nondecreasing. Second, for each fixed x, the sequence L * K (x) has an upper bound by conditions (ii) and (iii). Indeed, condition (ii) implies that Besides that, the function L * ∞ (x) is slowly varying. Let us prove the asymptotic relation which is analogous to (5). For all K ∈ N and x > 0, denote We have to prove the inequalities Condition (ii) implies that ). By Lemma 5, for all n ∈ N, we have F Sn ∈ R α and From these asymptotic relations we get that Using the obvious inequality lim inf x→∞ J lim inf x→∞ J 1 (K) and letting K tend to infinity, we derive from (10) the first inequality in (9). Since by condition (ii) of the theorem, we can use Lemma 6 for the numerator of J 2 (K) to obtain ∞ n=K+1 P(S n > x)P(η = n) c 2 F ξ1 (x) ∞ n=K+1 n p+1 P(η = n) with some positive constant c 2 . Therefore with some positive constant c 3 . The last inequality together with (10) implies that Letting K tend to infinity, we get the second desired inequality in (9) by condition (iii) of the theorem. This completes the proof of Theorem 7.
Proof of Theorem 8. Note that for all x > 0 and n ∈ N. Therefore with an arbitrary K 2.
For the first term, we have because lim x→∞ F ξ k (x)/F ξ1 (x) = d k for each fixed k.

Examples
In this section, we present two examples, which demonstrate the applicability of Theorem 7.
Note that this example deals with the same i.i.d. r.v.s as in Theorem 1. The difference is that Theorem 7 imposes stricter conditions on the c.r.v., which are sufficient for the d.f. of the random sum to be regularly varying as well as for real valued summands Example 2. Consider an example similar to that in [16]. Suppose that η is an arbitrary counting r.v. and {ξ 1 , ξ 2 , . . .} is a sequence of independent r.v.s distributed according to the two-sided Pareto laws where α > 0, and a − k , a + k , and b k are nonnegative constants such that a + k > 0 and a − k + b k + a + k 1 for all k ∈ N.
In this case, if lim sup n→∞ 1 n n k=1 a + k < ∞ and Eη 1+p < ∞ for some p > α, then conditions (i)-(iii) of Theorem 7 are satisfied, and Particularly, if η is distributed according to the Poisson law with parameter λ > 0 and a + k = 1/(k(k + 1)), k 1, then