On lower limits and equivalences for distribution tails of randomly stopped sums 1

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1 On lower limits and equivalences for distribution tails of randomly stopped sums 1 D. Denisov, 2 S. Foss, 3 and D. Korshunov 4 Eurandom, Heriot-Watt University and Sobolev Institute of Mathematics Abstract For a distribution F τ of a random sum S τ ξ ξ τ of i.i.d. random variables with a common distribution F on the half-line [, ), we study the limits of the ratios of tails F τ ()/F () as (here τ is an independent counting random variable). We also consider applications of obtained results to random walks, compound Poisson distributions, infinitely divisible laws, and sub-critical branching processes. AMS classification: Primary 6E5; secondary 6F1 Keywords: Convolution tail; Randomly stopped sums; Lower limit; Convolution equivalency; Subeponential distribution 1. Introduction. Let ξ 1, ξ 2,..., be independent identically distributed nonnegative random variables. We assume that their common distribution F on the half-line [, ) has an unbounded support, that is, F () F (, ) > for all. Put S and S n ξ ξ n, n 1, 2,.... Let τ be a counting random variable which does not depend on {ξ n } n 1 and has finite mean. Denote by F τ the distribution of a randomly stopped sum S τ ξ ξ τ. In this paper we discuss how does the tail behaviour of F τ relate to that of F and, in particular, under what conditions lim inf F τ () F () Eτ. (1) Relations on lower limits and on limits of ratios of from (1) have been first discussed by Rudin [21]. Theorem 2 of that paper states (for an integer p) the following Theorem 1. Let there eists a positive p [1, ) such that Eξ p, but Eτ p <. Then (1) holds. Rudin s studies were motivated by the paper [7] of Chover, Ney, and Wainger who considered, in particular, the problem of eistence of a limit for the ratio F τ () F () as. (2) From Theorem 1, it follows that, if F and τ satisfy its conditions and if a limit of (2) eists, then that limit must be equal Eτ. 1 The research of Denisov is partially supported by the Dutch BSIK project (BRICKS). The research of Foss and Korshunov supported by the Royal Society International Joint Project programme No.??? 2 Address: Eurandom, P.O. Bo MB Eindhoven, The Netherlands. address: Denisov@eurandom.tue.nl 3 Address: School of MACS, Heriot-Watt University, Edinburgh EH14 4AS, UK; and Sobolev Institute of Mathematics, 4 Koptyuga Pr., Novosibirsk 639, Russia. address: S.Foss@ma.hw.ac.uk 4 Address: Sobolev Institute of Mathematics, 4 Koptyuga pr., Novosibirsk 639, Russia. address: Korshunov@math.nsc.ru 1

2 Rudin proved Theorem 1 via probability generating functions techniques. Below we give an alternative and a more direct proof of Theorem 1 in the case of any positive p (i.e. not necessarily integer). Our method is based on truncation arguments; in this way, we propose a general scheme (see Theorem 4 below) which may be applied also to distributions with all finite moments. The condition Eξ p rules out a lot of distributions of interest, say, in the theory of subeponential distributions. For eample, log-normal and Weibull-type distributions have all moments finite. Our first result presents a natural moment condition on stopping time τ guaranteeing relation (1) for the whole class of heavy-tailed distributions. It is intuitively clear that, for that, τ should be light-tailed. Recall that a random variable ξ has a light-tailed distribution F on [, ) if Ee γξ < with some γ >. Otherwise F is called a heavy-tailed distribution; this happens if and only if Ee γξ for all γ >. Theorem 2. Let F be a heavy-tailed distribution and τ have a light-tailed distribution. Then (1) holds. Proof of Theorem 2 is based on a new technical tool (see Lemma 2) and significantly differs from a proof of Theorem 1 in [15] where a particular case τ 2 was considered. Theorem 2 is restricted to the case of light-tailed τ, but here etends Rudin s result to the class of all heavy-tailed distributions. The reasons for the restriction to Ee γτ < come from the proof of Theorem 2 but in fact are rather natural; the tail of τ should be lighter than the tail of any heavy-tailed distribution. Indeed, if ξ 1 1 then F τ () P{τ > }. This shows that the tail of F τ is at least as heavy as that of τ. Note that, in Theorem 1, in some sense, the tail of F τ is heavier than the tail of τ. Theorem 2 may be applied in various areas where randomly stopped sums do appear see Sections 8 11 (random walks, compound Poisson distributions, infinitely divisible laws, and branching processes) and, e.g., [17] for further eamples. For any distribution on [, ), let and ϕ(γ) e γ F (d) (, ], γ R, γ sup{γ : ϕ(γ) < } [, ]. Note that the moment-generating function ϕ(γ) is monotone continuous in the interval (, γ), and ϕ( γ) lim γ γ ϕ(γ) [1, ]. Theorem 3. Let ϕ( γ) < and E(ϕ( γ) + ε) τ < for some ε >. Assume that F τ () F () c as, where c (, ]. Then c E(τϕ τ 1 ( γ)). For (comments on) earlier partial results in the case τ 2, see, e.g., papers [6 8, 1, 15, 19, 2, 22] and further references therein. 2. Preliminary result. We start with the following Theorem 4. Let there eist a non-decreasing concave function h : R + R + such that Ee h(ξ) < and Eξe h(ξ). (3) 2

3 For any n 1, put A n Ee h(ξ ξ n). If F is heavy-tailed and if then (1) holds. EτA τ 1 <, (4) Proof. First we restate Theorem 1 of Rudin [21] in Lemma 1 below in terms of probability distributions and stopping times. Lemma 1. For any distribution F on [, ) with unbounded support and any independent counting random variable τ, lim inf F τ () F () It follows from Lemma 1 that it is sufficient to prove the upper bound in Theorem 4. Assume the contrary, i.e. there eist δ > and such that Eτ. F τ () (Eτ + δ)f () for all >. (5) For any positive b >, consider a concave function h b () min{h(), b}. (6) Since F is heavy-tailed, h() o() as. Therefore, for any fied b, there eists such that h b () h() for all >. Hence, by the condition (3), Ee h b(ξ) < and Eξe h b(ξ). (7) For any, we have the convergence h b () as b. Then, for any fied n, A n,b Ee h b(ξ ξ n) 1 as b. This and the condition (4) imply that there eists b such that EτA τ 1,b Eτ + δ/2. (8) For any random variable ζ and positive t, put ζ [t] min{ζ, t}. Then E(ξ [t] ξ[t] τ )e h b(ξ ξ τ ) by concavity of the function h b. Hence, E(ξ [t] ξ[t] τ )e h b(ξ ξ τ ) E(ξ [t] ξ[t] n )e h b(ξ ξ n) n Eξ[t] 1 eh b(ξ ξ n) P{τ n} n Eξ[t] +h b (ξ ξ n) n Eξ[t] Ee h b(ξ ξ n) na n 1,b P{τ n} P{τ n} P{τ n}, P{τ n} Eτ + δ/2, (9) 3

4 by (8). On the other hand, since (ξ ξ τ ) [t] ξ [t] ξ[t] τ, E(ξ [t] ξ[t] τ )e h b(ξ ξ τ ) The right side, after integration by parts, is equal to F τ ()d( [t] e hb() ) F ()d( [t] e hb() ). E(ξ ξ τ ) [t] e h b(ξ ξ τ ) [t] e hb() F τ (d) [t] e hb() F (d). (1) Since Eξ 1 e h b(ξ 1 ), both integrals (the divident and the divisor) in the latter fraction tend to infinity as t. For the non-decreasing function h b (), together with the assumption (5) it implies that lim inf t F τ ()d( [t] e hb() ) lim inf F τ ()d( [t] e hb() ) F ()d( [t] e hb() ) t F ()d( [t] e hb() ) Eτ + δ. Substituting this into (1) we get a contradiction to (9) for sufficiently large t. The proof is complete. 3. Proof of Theorem 1. Let an integer k be such that p 1 k < p. Without loss of generality, we can assume that Eξ k <. Consider a concave non-decreasing function h() (p 1) ln. Then Ee h(ξ 1) < and Eξ 1 e h(ξ 1). Thus, A n Ee h(ξ ξ n) E(ξ ξ n ) p 1 (E(ξ ξ n ) k ) (p 1)/k, since (p 1)/k 1. Further, where E(ξ ξ n ) k c n i 1,...,i k 1 cn k, E(ξ i1... ξ ik ) sup E(ξ i1... ξ ik ) <, 1 i 1,...,i k n due to Eξ k <. Hence, A n c (p 1)/k n p 1 for all n. Therefore, we get EτA τ 1 c (p 1)/k Eτ p <. All conditions of Theorem 4 are met and the proof is complete. 4. Characterization of heavy-tailed distributions. In the sequel we need the following eistence result which generalises a lemma by Rudin [21, page 989] onto the whole class of heavy-tailed distributions. Fi any δ (, 1]. Lemma 2. If a random variable ξ has a heavy-tailed distribution, then there eists a monotone concave function h : R + R + such that Ee h(ξ) 1 + δ and Eξe h(ξ). 4

5 Proof. Without loss of generality assume that ξ > a.s. We will construct a piecewise linear function h(). For that we introduce two sequences, n and ε n as n, and let h() h( n 1 ) + ε n ( n 1 ) if ( n 1, n ], n 1. This function is monotone, since ε n >. Moreover, this function is concave, due to the monotonicity of ε n. Put and h(). Since ξ is heavy-tailed, we can choose so that Choose ε 1 > so that which is equivalent to say that E{e ξ ; ξ (, 1 ]} + e 1 F ( 1 ) > e h( ) + δ F () + δ. E{e ε 1ξ ; ξ (, 1 ]} + e ε 1 1 F ( 1 ) e h( ) F () + δ/2, E{e h(ξ) ; ξ (, 1 ]} + e h( 1) F ( 1 ) e h( ) F () + δ/2, By induction we construct an increasing sequence n and a decreasing sequence ε n > such that n 2 n and E{e h(ξ) ; ξ ( n 1, n ]} + e h(n) F ( n ) e h( n 1) F ( n 1 ) + δ/2 n for any n 2. For n 1 this is already done. Make the induction hypothesis for some n 2. Due to heavy-tailedness, there eists n+1 2 n+1 so large that Note that E{e εn(ξ n) ; ξ ( n, n+1 ]} + e εn( n+1 n) F ( n+1 ) > 1 + δ. E{e ε n+1(ξ n) ; ξ ( n, n+1 ]} + e ε n+1( n+1 n) F ( n+1 ) as a function of ε n+1 is continuously decreasing to F ( n ) as ε n+1. Therefore, we can choose ε n+1 (, ε n ) so that E{e ε n+1(ξ n) ; ξ ( n, n+1 ]} + e ε n+1( n+1 n) F ( n+1 ) By definition of h() this is equivalent to the following equality: F ( n ) + δ/(2 n+1 e h(n) ). E{e h(ξ) ; ξ ( n, n+1 ]} + e h( n+1) F ( n+1 ) e h(n) F ( n ) + δ/2 n+1. Our induction hypothesis now holds with n + 1 in place of n as required. Net, Ee h(ξ) E{e h(ξ) ; ξ ( n 1, n ]} (e h(n 1) F ( n 1 ) e h(n) F ( n ) + δ/2 n) F () + δ 1 + δ. 5

6 On the other hand, since k 2 k, Then, for any n, E{ξe h(ξ) ; ξ > n } kn+1 2 n kn+1 2 n kn+1 E{ξe h(ξ) ; ξ ( k 1, k ]} E{e h(ξ) ; ξ ( k 1, k ]} ( e h( k 1) F ( k 1 ) e h( k) F ( k ) + δ/2 k). E{ξe h(ξ) ; ξ > n } 2 n e h(n) F ( n ) + δ δ, which implies Eξe h(ξ). Note also that necessarily lim n ε n ; otherwise lim inf h()/ > and ξ is light tailed. The proof of the lemma is complete. 5. Proof of Theorem 2. Since τ has a light-tailed distribution, Eτ(1 + ε) τ 1 < for some sufficiently small ε >. By Lemma 2, there eists a concave increasing function h, h(), such that Ee h(ξ 1) 1 + ε and Eξ 1 e h(ξ 1). Then by concavity A n Ee h(ξ ξ n) Ee h(ξ 1)+...+h(ξ n) (1 + ε) n. Combining altogether, we get EτA τ 1 <. All conditions of Theorem 4 are met and the proof is complete. 6. Fractional eponential moments. One can go further and obtain various results on lower limits and equivalencies for heavy-tailed distributions F which have all finite power moments (like Weibull and log-normal distributions). For instance, the following result takes place (see [9] for the proof): Let there eist α, < α < 1, such that Ee cξα for all c >. If Ee δτ α < for some δ >, then (1) holds. 7. Tail equivalence for randomly stopped sums. The following auiliary lemma compares the tail behavior of the convolution tail and that of the eponentially transformed distribution. Lemma 3. Let the distribution F and the number γ be such that ϕ(γ) <. Let the distribution G be the result of the eponential change of measure with parameter γ, i.e., G(du) e γu F (du)/ϕ(γ). Let τ be an independent stopping time such that Eϕ τ (γ) < and ν have the distribution P{ν k} ϕ k (γ)p{τ k}/eϕ τ (γ). Then lim inf G ν () G() 1 F Eϕ τ 1 lim inf τ () (γ) F () and lim sup G ν () G() 1 F Eϕ τ 1 lim sup τ () (γ) F (). 6

7 Proof. Put ĉ lim inf F τ () F (). By Lemma 1, ĉ [Eτ, ]. For any fied c (, ĉ), there eists > such that, for any >, By the total probability law, G ν () F τ () cf (). (11) P{ν k}g k () k1 k1 ϕ k (γ)p{τ k} Eϕ τ (γ) 1 Eϕ τ (γ) P{τ k} k1 After integration by parts, the latter sum is equal to [ P{τ k} e γ F k () + F k (y)de γy] k1 e γ F τ () + e γy F k (dy) ϕ k (γ) e γy F k (dy). F τ (y)de γy. Using also (11) we get, for >, G ν c [ () Eϕ τ e γ F () + F (y)de γy] (γ) c Eϕ τ e γy c F (dy) (γ) Eϕ τ 1 (γ) G(). Letting c ĉ, we obtain the first conclusion of the lemma. The proof of the second conclusion follows similarly. Lemma 4. If < γ <, ϕ( γ) <, and E(ϕ( γ) + ε) τ < for some ε >, then and lim inf lim sup F τ () F () F τ () F () Eτϕ τ 1 ( γ) Eτϕ τ 1 ( γ). Proof. We apply the eponential change of measure with parameter γ and consider the distribution G(du) e γu F (du)/ϕ( γ) and the stopping time ν with the distribution P{ν k} ϕ k ( γ)p{τ k}/eϕ τ ( γ). Again from the definition of γ, the distribution G is heavy-tailed. The distribution of ν is light-tailed, because Ee κν < with κ ln(ϕ( γ) + ε) ln ϕ( γ) >. Hence, lim sup G ν () G() lim inf G ν () G() Eν, by Theorem 2. The result now follows from Lemma 3 with γ γ, since Eν Eτϕ τ ( γ)/eϕ τ ( γ). 7

8 Proof of Theorem 3. In the case where F is heavy-tailed, we have γ and ϕ( γ) 1. By Theorem 2, c Eτ as required. In the case γ (, ) and ϕ( γ) <, the desired conclusion follows from Lemma Supremum of a random walk. Let {ξ n } be a sequence of independent random variables with a common distribution F on R and Eξ 1 m <. Put S, S n ξ ξ n. By the SLLN, M sup n S n is finite with probability 1. Let F I be the integrated-tail distribution on R +, that is, ( ) F I () min 1, F (y)dy, >. It is well-known (see, e.g. [1, 12, 13] and references therein) that if F I S, then P{M > } 1 m F I () as. (12) Korshunov [18] proved the converse: (12) implies F I S. Now we accompany this assertion by the following Theorem 5. Let F I be long-tailed, that is, F I ( + 1) F I () as. If, for some c >, then c 1/m and F I is subeponential. Proof. Consider the defective stopping time P{M > } cf I () as, η inf{n 1 : S n > } and let {ψ n } be i.i.d. random variables with common distribution function G() P{ψ n } P{S η η < }. It is well-known (see, e.g. Feller [14, Chapter 12]) that the distribution of the maimum M coincides with the distribution of the randomly stopped sum ψ ψ τ, where the stopping time τ is independent of the sequence {ψ n } and is geometrically distributed with parameter p P{M > } < 1, i.e., P{τ k} (1 p)p k for k, 1,.... Equivalently, P{M B} G τ (B). From Borovkov [4, Chapter 4, Theorem 1], if F I is long-tailed, then Then it follows from the theorem hypothesis that G() 1 p pm F I (). (13) G τ () cpm G() as. 1 p Therefore, by Theorem 3 with γ, c Eτ(1 p)/pm 1/m. Then it follows from [11] that F I is subeponential. Now the proof is complete. 9. The compound Poisson distribution. Let F be a distribution on R + and t a positive constant. Let G be the compound Poisson distribution G e t n t n n! F n. Considering τ in Theorem 3 with P{τ n} t n e t /n!, we get 8

9 Theorem 6. Let ϕ( γ) <. If, for some c >, G() cf () as, then c te t(ϕ( γ) 1). Corollary 1. The following statements are equivalent: (i) F is subeponential; (ii) G is subeponential; (iii) G() tf () as ; (iv) F is heavy-tailed and G() cf () as, for some c >. Proof. Equivalence of (i), (ii), and (iii) was proved in [11, Theorem 3]. The implication (iv) (iii) follows from Theorem 3 with γ. Some local aspects of this problem for heavy-tailed distributions were discussed in [2, Theorem 6]. 1. Infinitely divisible laws. Let F be an infinitely divisible law on [, ). The Laplace transform of an infinitely divisible law F can be epressed as e λ F (d) e aλ (1 e λ )ν(d) (see, e.g. [14, Chapter XVII]). Here a is a constant and the Lévy measure ν is a Borel measure on (, ) with the properties µ ν(1, ) < and 1 ν(d) <. Put G(B) ν(b (1, ))/µ. The relations between the tail behaviour of measure F and the corresponding Lévy measure ν were considered in [11, 19]. The local analogue of that result was proved in [2]. We strength the corresponding result of [11] in the following way. Theorem 7. The following assertions are equivalent: (i) F is subeponential; (ii) G is subeponential; (iii) ν() F () as ; (iv) F is heavy-tailed and ν() cf () as, for some c >. Proof. Equivalence of (i), (ii), and (iii) was proved in [11, Theorem 1]. It remains to prove the implication (iv) (iii). It is pointed out in [11] that the distribution F admits the representation F F 1 F 2, where F 1 () O(e ε ) for some ε > and F 2 (B) e µ n µ n n! G n (B). Since F is heavy-tailed and F 1 is light-tailed, we get the equivalence F () F 2 () as. Therefore, as, µg() ν() cf () cf 2 (). With necessity G is heavy-tailed, and c 1 by Corollary Branching processes. In this section we consider the limit behaviour of sub-critical, age-dependent branching processes for which the Malthusian parameter does not eist. Let h(z) be the particle production generating function of an age-dependent branching process with particle lifetime distribution F (see [3, Chapter IV], [16, Chapter VI] for background). We take the process to be sub-critical, i.e. A h (1) < 1. Let Z(t) denote the number of particles 9

10 at time t. It is known (see, for eample, [3, Chapter IV, Section 5] or [5]) that EZ(t) admits the representation EZ(t) (1 A) A n 1 F n (t). It was proved in [5] for sufficiently small values of A and then in [6, 7] for any A < 1 that EZ(t) F (t)/(1 A) as t, provided F is subeponential. The local asymptotics were considered in [2]. Applying Theorem 3 with τ geometrically distributed and γ, we deduce Theorem 8. Let F be heavy-tailed, and, for some c >, EZ(t) cf (t) as t. Then c 1/(1 A) and F is subeponential. References 1. Asmussen, S., 2. Ruin Probabilities, World Scientific, Singapore. 2. Asmussen S., Foss S., Korshunov D., 23. Asymptotics for sums of random variables with local subeponential behaviour. J. Theoret. Probab. 16, Athreya, K., and Ney, P., Branching Processes, Springer, Berlin. 4. Borovkov, A. A., Stochastic Processes in Queueing Theory, Springer, Berlin. 5. Chistyakov, V. P., A theorem on sums of independent random positive variables and its applications to branching processes. Theory Probab. Appl. 9, Chover, J., Ney, P. and Wainger, S., Functions of probability measures. J. Anal. Math. 26, Chover J., Ney P., and Wainger S., Degeneracy properties of subcritical branching processes. Ann. Probab. 1, Cline, D., Convolutions of distributions with eponential and subeponential tailes. J. Aust. Math. Soc. 43, Denisov, D., Foss, S. and Korshunov, D. Asymptotic properties of randomly stopped sums in the presence of heavy tails (working paper). 1. Embrechts, P. and Goldie, C. M., On convolution tails. Stochastic Process. Appl. 13, Embrechts, P., Goldie, C. M., and Veraverbeke, N., Subeponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Etremal Events for Insurance and Finance, Springer, Berlin. 13. Embrechts, P., and Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math. and Economics 1, Feller, W., An Introduction to Probability Theory and Its Applications Vol. 2, Wiley, New York. 15. Foss, S. and Korshunov, D., 27. Lower limits and equivalences for convolution tails. Ann. Probab. 35, No Harris, T., The Theory of Branching Processes, Springer, Berlin. 17. Kalashnikov, V., Geometric sums: bounds for rare events with applications. Risk analysis, reliability, queueing., Mathematics and its Applications, 413. Kluwer, Dordrecht. 1

11 18. Korshunov, D., On distribution tail of the maimum of a random walk. Stoch. Proc. Appl. 72, Pakes, A. G., 24. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41, Rogozin, B. A., 2. On the constant in the definition of subeponential distributions. Theory Probab. Appl. 44, Rudin, W., Limits of ratios of tails of measures. Ann. Probab. 1, Teugels, J. L., The class of subeponential distributions. Ann. Probab. 3,