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ON TAILS OF PERPETUITIES PAWEL HITCZENKO† 1 Abstract. We establish an upper bound on the tails of a random variable that arises 1 as a solution of a stochastic difference equation. In the non–negative case our bound is 0 similar to a lower bound obtained by Goldie and Gru¨bel in 1996. 2 n a J 1. Introduction 1 1 A random variable R satisfying the distributional identity ] R (1) R =d MR+Q, P . where (M,Q) are independent of R on the right-hand side and =d denotes the equality in h t distribution, is referred to as perpetuity and plays an important role in applied probability. a m The main reason for this is that it appears as a limit in distribution of a sequence (R ) n [ given by d 1 R = M R +Q , n ≥ 1, n n n−1 n v 3 provided that limit exists (here, (Mn,Qn) is a sequence of i.i.d. random vectors distributed 8 like (M,Q) and R could be an arbitrary random variable; for convenience we will set 0 1 R = 0). Systematic study of properties of such sequences was initiated by Kesten in [5] 2 0 . and they continue till this day. Once the convergence in distribution of (R ) is established, 1 n 0 at the center ofthe investigation is the tail behavior of R. There aretwo distinctly different 1 cases: 1 : P(|M| > 1) > 0 and P(|M| ≤ 1) = 1. v i The first results in R having a heavy tail distribution, that is X r P(|R| > x) ∼ Cx−κ, a for a suitably chosen constant κ and some constant C (see the original paper of Kesten [5] or [2]), while in the second case the tails of R are no heavier than exponential. This was observed by Goldie and Gru¨bel in [3]. Some subsequent work is in [4], but the full picture in this case is not complete. The purpose of this note is to shed some additional light on that case by establishing a universal upper bound on the tails of |R|. In a special, but important, situation when Q and M (and thus also R) are non–negative our bound is comparable to a lower bound obtained by Goldie and Gru¨bel in [3]. 1991 Mathematics Subject Classification. 60E15,60H25. Key words and phrases. perpetuity, stochastic difference equation, tail behavior. † Supported in part by the NSA grant #H98230-09-1-0062. 1 2 PAWEL HITCZENKO 2. Bounds on the tails For a random variable M such that |M| ≤ 1 and 0 < δ < 1 define p := P(1 − δ ≤ δ |M| ≤ 1). Then, as has been shown in [3] (see also the equation (2.2) in [4]) if 0 ≤ M ≤ 1 and Q ≡ q (q being a positive constant), then for 0 < c < 1 and x > q we have ln(1−c) P(R > x) ≥ exp( lnp ). cq/x ln(1−cq/x) Since ln(1− cq/x) ≤ −cq/x, for any particular value of c, say c = 1/2, this immediately gives ln(1−c) 2ln2 P(R > x) ≥ exp(− xln(p )) = exp( xlnp ). cq/x q/(2x) cq q Our aim here is to supply an upper bound of a similar form. While our result does not give the asymptotics of P(R > x) as x → ∞, it shows that it essentially behaves like exp(c1xlnp ) for some positive constants c ,c . Specifically, we prove q c2q/x 1 2 Proposition 1. Assume |Q| ≤ q, |M| ≤ 1, and let R be given by (1). Then, for sufficiently large x 1 P(|R| > x) ≤ exp( xlnp ). 2q/x 4q Thus, if Q ≡ q > 0 and 0 ≤ M ≤ 1 then 2ln2 1 exp( xlnp ) ≤ P(R > x) ≤ exp( xlnp ). q/(2x) 2q/x q 4q Proof. If P(|M| = 1) > 0 then, as was proved in [3], R has tails bounded by those of an exponential variable, so we assume that |M| has no atom at 1. Fix 0 < δ < 1 and define a sequence (T ) as follows k T = 0, T = inf{k ≥ 1 : |M | ≤ 1−δ}, m ≥ 1. 0 m Tm−1+k Then T ’s are i.i.d. random variables, each having a geometric distibution with parameter k 1 − p . Furthermore, |M | ≤ 1 − δ if k = T + ··· + T for some i ≥ 1 and |M | ≤ 1 δ k 1 i k otherwise. Therefore, m Y|Mk| ≤ (1−δ)j for T1 +···+Tj ≤ m < T1 +···+Tj +Tj+1. k=1 This in turn implies that k−1 k−1 (cid:12)(cid:12)XYMj(cid:12)(cid:12) ≤ XY|Mj| ≤ T1 +(1−δ)T2 +(1−δ)2T3 +··· = X(1−δ)k−1Tk. (cid:12)k≥1 j=1 (cid:12) k≥1 j=1 k≥1 Therefore, if |Q| ≤ q we get k−1 x x (2) P(|R| > x) ≤ P(XY|Mj| ≥ ) ≤ P(XTk(1−δ)k−1 ≥ ). q q k≥1 j=1 k≥1 ON TAILS OF PERPETUITIES 3 To bound the latter probability we use a widely known argument (our calculations follow [1, proof of Proposition 2]). First, if T is a geometric variable with parameter 1−p then ∞ ∞ eλ(1−p) eλ EeλT = XeλjP(T = j) = Xeλjpj−1(1−p) = = , 1−eλp 1− p (eλ −1) j=1 j=1 1−p provided eλp < 1. Thus, writing t in place of x/q in the right-hand side of (2), for λ > 0 we have P(X(1−δ)k−1Tk ≥ t) = P(exp(λX(1−δ)k−1Tk) ≥ eλt) ≤ e−λtEeλPk≥1Tk(1−δ)k−1. k≥1 k≥1 If λ satisfies eλp < 1 then peλ(1−δ)k−1 < 1 for every k ≥ 1 as well, and by independence of (T ), the expectation on the right is k ∞ eλ(1−δ)k−1 ∞ 1 (3) Y = eλ/δY . 1− p (eλ(1−δ)k−1 −1) 1− p (eλ(1−δ)k−1 −1) k=1 1−p k=1 1−p Now, choose λ > 0 so that p (eλ−1) ≤ 1. Then, as 1/(1−u) ≤ e2u for 0 ≤ u ≤ 1/2, for 1−p 2 every k ≥ 1 we get 1 p ≤ exp(2 (eλ(1−δ)k−1 −1)). 1− p (eλ(1−δ)k−1 −1) 1−p 1−p Therefore, the rightmost product in (3) is bounded by p exp(2 X(eλ(1−δ)k−1) −1)). 1−p k≥1 We bound the sum in the exponent as follows λj(1−δ)(k−1)j λj XX = X X(1−δ)j(k−1) j! j! k≥1 j≥1 j≥1 k≥1 λj 1 1 λj eλ −1 = X ≤ X = . j! 1−(1−δ)j δ j! δ j≥1 j≥1 Combining the above estimates we get that λ 2p eλ −1 (4) P(|R| > qt) ≤ exp(−tλ+ + ). δ 1−p δ provided that λ satisfies the required conditions, that is: p 1 eλp < 1 and (eλ −1) ≤ . 1−p 2 Clearly both are satisfied when eλp ≤ 1/2. We finish the proof by making a suitable choice of λ. Since we are assuming that |M| has no atom at 1 and we are interested in large x, we may assume that δ is small enough 4 PAWEL HITCZENKO so that p < 1/3. This condition implies that 2p /(1−p ) < 3p so that the last term in δ δ δ δ the exponent of (4) is bounded by 3p (eλ −1)/δ. Now let t = 2/δ. Then (4) becomes δ 2 λ 2p eλ −1 1 P(|R| > qt) ≤ exp(−λ + + δ ) ≤ exp(− (λ−3p (eλ −1))). δ δ δ 1−p δ δ δ Set λ = ln( 1 ) so that eλp = 1. This choice of λ is within the constraints and maximizes 3pδ δ 3 the value of λ−3p (eλ −1), this maximal value being δ 1 1 1 1 ln( )−3p ( −1) = ln( )−(1+ln3)+3p ≥ ln(1/p ), δ δ δ 3p 3p p 2 δ δ δ with the inequality valid for sufficiently small p (less than e−2/9 for example). Thus, using δ t = 2/δ we finally obtain 1 t P(|R| > qt) ≤ exp(− ln(1/p )) = exp( lnp ), δ 2/t 2δ 4 or, in terms of x, x P(|R| > x) ≤ exp( lnp ). 2q/x 4q (cid:3) References [1] Goh, W. M. Y. and Hitczenko, P. (2008).Randompartitions with restrictedpartsizes. Random Structures Algorithms. 32, 440–462. [2] Goldie, C. M.(1991).Implicitrenewaltheoryandtailsofsolutionsofrandomequations.Ann.Appl. Probab. 1, 126–166. [3] Goldie, C. M. and Gru¨bel, R. (1996). Perpetuities with thin tails. Adv. in Appl. Probab. 28, 463–480. [4] Hitczenko, P. and Weso lowski, J. (2009). Perpetuities with thin tails, revisited. Ann. Appl. Probab. 19, 2080 – 2101. (Corrected version available at http://arxiv.org/abs/0912.1694.) [5] Kesten,H.(1973).Randomdifferenceequationsandrenewaltheoryforproductsofrandommatrices. Acta Math. 131, 207–248. Pawe l Hitczenko, Departments of Mathematics and Computer Science, Drexel Univer- sity, Philadelphia, PA 19104, U.S.A E-mail address: [email protected] URL: http://www.math.drexel.edu/∼phitczen