Randomly stopped minima and maxima with exponential-type distributions

Abstract. Let {ξ1, ξ2, . . . } be a sequence of independent real-valued and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at 0 and integer-valued random variable, which is independent of {ξ1, ξ2, . . . }. In this paper, we consider conditions for {ξ1, ξ2, . . . } and η under which the distributions of the randomly stopped maxima and minima, as well as randomly stopped maxima of sums and randomly stopped minima of sums, belong to the class of exponential distributions.

• For any γ 0, we denote by L(γ) the class of exponential-type d.f.s.It is said that = e −γy for any y > 0.
• For γ = 0, the class L(0) is called the class of long-tailed distributions and denoted by L. Consequently, F ∈ L if and only if By Proposition 2.6 from [2], an absolutely continuous d.f.F belongs to the class L(γ), γ 0, if and only if for some measurable functions α and β with α(x) + β(x) 0 for all x ∈ R such that Another representation formula, which does not require the absolute continuity of F , can be found in [27].It states that F ∈ L(γ) with γ 0 if and only if for some positive measurable functions ᾱ and β such that We note here that any gamma distribution belongs to the class L(γ) with some γ > 0. In particular, any exponential or Erlang distribution belongs to this class.
In addition, a d.f.from the class L(γ) can be stepped, i.e. there is such a function that describes a distribution of a discrete r.v.To be more precise, if γ > 0, we define F (x) = exp(−γ log e x ) for all x 0, where a denotes the integer part of a real number a.It is obvious that this function F is a d.f. of a nonnegative discrete r.v.Moreover, F ∈ L(γ) because for any y > 0, we have The class of long-tailed distributions L (but not the term itself) was introduced by Chistyakov [7] in the context of branching processes, whereas for γ > 0, the class L(γ) was introduced by Chover et al. [8,9].In this paper, we suppose that the d.f.s of the r.v.s under consideration belong to either of the following four classes: A number of interesting and important properties of distributions from these classes can be found in the book by Foss et al. [19] and in the papers by Albin and Sundén [2], Beck et al. [4], Cheng et al. [6], Chover et al. [8,9], Cline [10,11], Cui et al. [12], Embrechts and Goldie [18], Klüppelberg [20], Omey et al. [23], Pakes [24,25], Shimura and Watanabe [26], Watanabe [28], Watanabe and Yamamuro [29] and Xu et al. [30][31][32], Yang et al. [33], among others.
A similar assertion but for the case γ = 0 was obtained by Leipus and Šiaulys (see [21,Thm. 6]).In such a case, the restriction on the c.r.v.η is substantially weaker.We present their result below.Theorem 2. Let {ξ 1 , ξ 2 , . ..} be a sequence of i.i.d.r.v.s distributed on R with d.f.F ξ .If F ξ ∈ L, then F Sη belongs to the class L for any c.r.v.η satisfying the condition for each δ ∈ (0, 1).
In the original paper, the assertion of the theorem above is formulated for nonnegative r.v.s only.But it is easy to check that the proof is identical for a more general situation.Also we should note that d.f.F Sη can be exponential without requirement of exponentiality for F ξ .The following assertion was proved by Xu et If the sequence of r.v.s {ξ 1 , ξ 2 , . ..} consists of independent but not necessarily identically distributed r.v.s, then the following generalization of Theorem 2 was obtained by Danilenko et al. (see [13,Thm. 4]).
The rest of the paper is organized as follows.In Section 2, we present our main results.Section 3 consists of two 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 assertions below, we suppose that the sequence {ξ 1 , ξ 2 , . . .} consists of independent r.v.s and the c.r.v.η is independent of this sequence.By default, we suppose also that r.v.s ξ 1 , ξ 2 , . . .are distributed on R, i.e. they can take positive and negative values.
The first theorem describes properties of the randomly stopped minima.
Theorem 5.For the randomly stopped minima, the following assertions hold: The second theorem describes properties of the randomly stopped maxima.We note that it gives only sufficient conditions.Theorem 6.Let B denote one of d.f.s classes: κ and some κ ∈ supp η := {n: P(η = n) > 0}, and there is a positive sequence ϕ(n), n ∈ N, such that Nonlinear Anal.Model.Control, 24(2):297-313 The following assertion gives sufficient conditions under which the randomly stopped minima of sums preserves exponentiality.
Theorem 7. Suppose that B denotes the same object as in Theorem 6.For the randomly stopped minima of sums, the following two assertions hold: The last assertion describes sufficient conditions under which the d.f. of the randomly stopped maxima of sums remains in the class of exponential distributions.We note that Theorem 8 below is related to the results of the paper by Danilenko et al. [13], where the authors consider conditions under which the randomly stopped sums preserve exponentiality.
Theorem 8. Suppose that B denotes the same object as in Theorem 6.Then the following assertions hold: for some γ 0 and each y 0, then F S (η) ∈ L(γ) for any c.r.v.satisfying condition (3).

Auxiliary lemmas
In this section, we give two auxiliary assertions, which are used in the proofs of our main results.The first lemma is an extension of Lemma 3.1 from [17].
Lemma 1.Let X and Y be two independent r.v.s with d.f.s F and G, respectively, and let H be the d.f. of max(X, Y ).Then if F (x) > 0, G(x) > 0 and t 0.
In addition, Proof.The derivation of the upper bound can be found in the proof of Lemma 3.1 in [17].
The next lemma was proved by Embrechts and Goldie (see [18,Thm. 3]).Note that in its assertion " * " stands for the convolution of d.f.s.

Proofs of main results
In this section, we give detailed proofs of all our main results.
Then for any x > 0, we have Therefore, we obtain . .for some positive parameters γ 1 , γ 2 , . . . .Hence, for the finite nonrandom κ because for each y > 0. Thus, it follows from ( 6) and ( 7) that for any c.r.v.η, which proves part (i) of the theorem.
Part (ii) Let us consider the class L ∞ .If F ξ k ∈ L ∞ for each k ∈ N, then, applying arguments similar to those in the proof of part (i), we deduce that F ξ (η) ∈ L ∞ for an arbitrary c.r.v.η.
If F ξ (η) ∈ L ∞ for an arbitrary c.r.v.η, then from ( 6) it follows that F ξ (m) ∈ L ∞ for any m ∈ N.This implies that F ξ k ∈ L ∞ for each k ∈ N because for all x > 0 and y > 0, we have and which completes the proof of part (ii).If F ξ k ∈ L for each index k ∈ N, then proof of the assertion is analogous to the presented proof for the class L ∞ .
Proof of Theorem 6.We consider the proof separately for different classes.
• Let us consider the case B = L(γ) with some γ 0. First, we suppose that P(η < κ) > 0. Let y > 0 be a real number.By (5), for any x > y and K > κ, we have where .
Applying arguments similar to those for deriving (8), we obtain where Similarly to (10), we get Applying the lower bound from Lemma 1, we obtain Therefore, by ( 5), we have Combining inequalities ( 12), ( 13) and ( 14) and taking into account conditions (4), we deduce that The last inequality together with (11) implies that F ξ (η) ∈ L(γ), which is the desired conclusion for the case P(η < κ) > 0. If P(η < κ) = 0, then the proof is similar with the only difference that for all x > 0. This completes the proof provided that B = L(γ) with some γ 0.
Applying arguments similar to those for deriving (8), ( 10), ( 12) and ( 13), we have https://www.mii.vu.lt/NA and for all y > 0, K > κ and sufficiently large x, where The last two inequalities, condition (4) and relation ( 16) yield If either P(η < κ) = 0 or κ = 1, then we have I 1 (x) = 0 for all x > 0. Therefore, from the last two inequalities and condition (4) we see that then the proof can be obtained along the lines of the proof for the class L ∞ + .
Proof of Theorem 7. We give the proof only for the case B = L(γ) with some γ 0. The other two cases can be considered similarly.
Applying arguments similar to those for deriving (17), we obtain for all x > 0. From this and inequality (5) it follows that for all x > 0 and y > 0. Letting x → ∞ in the last inequality, we see that Continuing in the same way, we deduce that F S (n) ∈ L(γ) for each n ∈ N.
Part (i) In the case under consideration, we have Applying (5) yields for all x > 0 and y > 0. Since F S (n) ∈ L(γ) for each n, the last inequality implies that F S (η) ∈ L(γ) as well, which proves part (i) of the theorem.
Part (ii) It is easily seen that the condition on the c.r.v.η implies that P(η = k) > 0 for all k large enough.In addition, we observe that requirement (3) implies relation which follows from the equality provided for k is sufficiently large. https://www.mii.vu.lt/NA We now choose any K large enough.Then on the one hand, we have for all x > 0 because for any n ∈ {K, K + 1, . ..} in the case under consideration.On the other hand, it is evident that Therefore, by inequality (5), for all x > 0 and y > 0, we obtain for any fixed y > 0 and all K large enough.Similarly, we get for any fixed y > 0 and all K large enough.The last two inequalities together with (20) imply that F S (η) ∈ L(γ) for any c.r.v.η satisfying property (3), which proves part (ii) of the theorem.
Proof of Theorem 8. Part (i) We consider the proof separately for different classes.
• Let now B = L(γ) with some γ 0. From Lemma 2 it may be concluded that the d.f.s of the r.v.s belong to the class L(γ) for any n κ, where ξ + k denotes the positive part of ξ k , 1 k κ.Moreover, it is obvious that for any x > 0 and n κ.Consequently, F S (n) ∈ L(γ) for each n κ.
To get the desired assertion, it suffices to notice that and apply arguments similar to those in the proof of part (i) of Theorem 7.
• The case of the class L ∞ can be considered similarly, which proves part (i) of the theorem.
Part (ii) This part of the theorem follows immediately from Theorem 4 because distributions of the r.v.s S η and S (η) coincide provided that all the r.v.s in the sequence {ξ 1 , ξ 2 , . ..} are nonnegative.This completes the proof of the theorem.

Examples
In this section, we present two examples which demonstrate the applicability of the obtained results.