Limit theorems for discounted convergent perpetuities

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for convergent perpetuities $\sum_{k\geq 0}b^{\xi_1+\ldots+\xi_k}\eta_{k+1}$ as $b\to 1-$. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.


Introduction
Let (ξ 1 , η 1 ), (ξ 2 , η 2 ), . . . be independent copies of an R 2 -valued random vector (ξ, η) with arbitrarily dependent components. Denote by (S k ) k∈N 0 (as usual, N 0 := N ∪ {0}) the standard random walk with jumps ξ k defined by S 0 := 0 and S k := ξ 1 + . . . + ξ k for k ∈ N. Whenever a random series k≥0 e −S k η k+1 converges a.s., its sum is called perpetuity because of the following actuarial application. Assuming, for the time being, that ξ and η are a.s. positive, we can interpret η k and e −ξ k as the planned payment and the discount factor (risk) for year k, respectively. Then k≥0 e −S k η k+1 can be thought of as 'the present value of a permanent commitment to make a payment ... annually into the future forever' (the phrase borrowed from p. 1196 in [13]). When studying the aforementioned random series from purely mathematical viewpoint, the one-sided assumptions are normally omitted whereas the term 'perpetuity' is still used. See the books [7] and [16] for surveys of the area of perpetuities from two different perspectives.
In the present paper we investigate the asymptotic behavior as b → 1− of the convergent series k≥0 b S k η k+1 that we call discounted convergent perpetuity. We intend to prove the basic limit theorems for the discounted convergent perpetuities: a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm. Getting back to the actuarial interpretation, these results describe the fluctuations of the present value when the actuarial market is close to the customer-friendly scenario of no risk.
A sufficient condition for the almost sure (a.s.) absolute convergence of the random series k≥0 b S k η k+1 with fixed b ∈ (0, 1) is Eξ ∈ (0, ∞) and E log + |η| < ∞, see, for instance, Theorem 2.1 in [13]. This sufficient condition holds, that is, the discounted perpetuity is well-defined for all b ∈ (0, 1), under the assumptions of all our results to be formulated soon.
Throughout the paper we write P → to denote convergence in probability, and ⇒ and d −→ to denote weak convergence in a function space and weak convergence of one-dimensional distributions, respectively. Also, we denote by D(0, ∞) the Skorokhod space of right-continuous functions defined on (0, ∞) with finite limits from the left at positive points. We proceed by giving a functional central limit theorem. Theorem 1.2. Assume that µ = Eξ ∈ (0, ∞), Eη = 0 and s 2 := Var η ∈ (0, ∞). Then, as b → 1−, e −uy dB(y) u>0 (2) in the J 1 -topology on D(0, ∞), where (B(t)) t≥0 is a standard Brownian motion.
Remark 1.3. The limit process in Theorem 1.2 is an a.s. continuous Gaussian process on (0, ∞) with covariance Such a process has appeared in the recent articles [8], [17] and [18]. The latter paper provides additional references.
Putting in (2) u = 1 and using (3) with u = v = 1 we obtain a one-dimensional central limit theorem.
where Normal(0, 1) denotes a random variable with the standard normal distribution.
Finally, we are interested in the rate of a.s. convergence in Theorem 1.1 when m = 0 which is expressed by a law of the iterated logarithm. A hint concerning the form of this law is given by the central limit theorem, Corollary 1.4. For a family (x t ) we denote by C((x t )) the set of its limit points. Theorem 1.5. Assume that µ = Eξ ∈ (0, ∞), Eη = 0 and s 2 = Var η ∈ (0, ∞). Then lim sup (lim inf) b→1− 1 − b 2 log log 1 In particular,

Related literature
Random power series. The random power (or geometric) series k≥0 b k η k+1 for b ∈ (0, 1) is a rather particular case of a discounted convergent perpetuity which corresponds to the degenerate random walk S k = k for k ∈ N 0 . In this section we first discuss known counterparts of our main results for the random power series.
Law of large numbers. Under the assumption E|η| < ∞, the following strong law of large numbers can be found in Theorem 1 of [19] lim b→1− where m = Eη. Central limit theorem. Under the assumption E|η| 3 < ∞ Theorem 1 in [12] proves a Berry-Esséen inequality which entails where s 2 = Var η ∈ (0, ∞). Theorem 4.1 in [25] is a functional limit theorem in the Skorokhod space for the process ( , properly normalized and centered, as b → 1−. Here and hereafter, ⌊x⌋ denotes the integer part of real x. The corresponding limit process is a time-changed Brownian motion. Law of iterated logarithm. It was proved in Theorem 3 of [11] that for centered bounded η k with variance s 2 . In Theorem 2 of [19] this limit relation was stated without proof, for not necessarily bounded η k . Our Theorem 1.5 is an analogue of Theorem 1.1 in [5] dealing with the random power series. In Theorem 1.1 of [22] the sequence (η k ) k∈N is stationary, conditionally centered and ergodic with Eη 2 1 < ∞. In this more general setting the authors prove a counterpart of (4) for the corresponding random power series. Another proof in both settings based on a strong approximation result is given in Theorem 2.1 of [26]. See also [10] and [23] for related results.
Although the random power series is a toy example of perpetuities, transferring results from the former to the latter may be a challenge. To justify this claim, we only mention that while necessary and sufficient conditions for the a.s. convergence of random power series can be easily obtained (just use the Cauchy root test in combination with the Borel-Cantelli lemma), the corresponding result for perpetuities is highly non-trivial, see Theorem 2.1 in [13] and its proof. The reason is clear: the random power series is a weighted sum of independent random variables, whereas it is not the case for perpetuities.
Investigation of (general) weighted sums of independent identically distributed random variables has been and still is a rather popular trend of research. We refrain from giving a survey and only mention recent contributions [1,2] in which a random Dirichlet series is analyzed. Discounted perpetuities. As far as we know, Theorems 1.1, 1.2 and 1.5 are new. Under the additional assumption Eξ 2 < ∞ (we only require Eξ ∈ (0, ∞)) our Corollary 1.4 follows from Theorem 6.1 in [24] which we state as Proposition 2.1 for reader's convenience.
We stress that our idea of proof of Theorem 1.2 is different from Vervaat's. Also, we note that in Theorem 2 of [9] the method of moments is employed for proving a (one-dimensional) central limit theorem for k≥0 b S k as b → 1− under the assumptions ξ ≥ 0 a.s. and Eξ p < ∞ for all p > 0.
3 Proof of Theorem 1.1 We shall use a fragment of Theorem 5 on p. 49 in [14] that we give in a form adapted to our setting.
Lemma 3.1. Let (c k (b)) k∈N and (s k ) k∈N be sequences of real-valued functions defined on (0, 1) and real numbers, respectively. Assume that (i) k≥1 |c k (b)| < ∞ for all b ∈ (0, 1) and that, for some b 0 ∈ (0, 1) and some A > 0 which does Proof of Theorem 1.1. We first prove that For x ∈ R, put M (x) = #{n ≥ 0 : S n ≤ x}. Since lim n→∞ S n = +∞ a.s., we have M (x) < ∞ a.s. Furthermore, by Theorem B in [20], lim x→∞ x −1 M (x) = µ −1 a.s. Hence, given ε > 0 there exists an a.s. finite The number of summands in the sum on the right-hand side is a.s. finite, for it is equal to M (x 0 ),

s. Integration by parts yields
Thus, The proof of the converse inequality for the limit inferior is completely analogous.
Passing to the proof of (1) we use summation by parts to obtain, for b ∈ (0, 1) and ℓ ∈ N, where T 0 := 0 and T k := η 1 + . . . + η k for k ∈ N. We have lim ℓ→∞ b S ℓ−1 T ℓ = 0 a.s. because by the strong law of large numbers the first factor decreases to zero exponentially fast, whereas the second factor exhibits at most linear growth. Hence, We are going to apply Lemma 3.
Later on, we shall need the following result. Its proof is omitted, for it is analogous to the proof of Theorem 1.1. y λk n η k = me −a a.s.
Clearly, these limit relations also hold if we put formally x n = y n = b and let b → 1−, that is, if one passes to the limit continuously.

Proof of Theorem 1.2
We shall prove weak convergence of the finite-dimensional distributions and then tightness.
Passing to the proof of (11) we first conclude that, in view of which holds for a 1 , . . . , a ℓ ∈ R and y > 0, it suffices to show that, for all ε > 0 and u > 0, Put T := sup{n ∈ N 0 : S n ≤ 0} and note that T < ∞ a.s. as a consequence of lim n→∞ S n = +∞ a.s. We infer To proceed, observe that, for k ≥ T + 1, we have b uS k ≤ 1, whence This yields The limit relation is justified by the fact that while the truncated second moment converges to For the proof of Proposition 5.7 we need the following one-dimensional central limit theorem.
After noting that k=0 b 2µk a.s. as b → 1− by the strong law of large numbers for random walks, a simplified version of the proof given above applies. We omit details.
Here, each summand converges a.s. as b → 1− to an a.s. finite random variable. Furthermore, the number of nonzero summands is a.s. finite in view of consequence of the strong law of large numbers. Thus, (12) has been proved. Next, we intend to show that, for any u, v ∈ [c, d] and b < 1 close to 1, for a constant A which does not depend on u and v. Here, R c k (δ) denotes the complement of To this end, we observe that R c k (δ) ⊆ {S k > 0} and then invoking the mean value theorem for differentiable functions we obtain a.s. on R c k (δ) We have used the inequality sup x>0 | log b|xb x ≤ 1/e for the last step. It remains to note that Thus, (13) are tight. The proof of Theorem 1.2 is complete.

Proof of Theorem 1.5
Our argument follows closely the paths of (slightly different) proofs of Theorem 1.1 in [5] and Theorem 1.1 in [22]. In the cited references S n = n, n ∈ N 0 , that is, the random walk (S n ) n∈N 0 is deterministic. Of course, we know that in our setting, for large n, S n is approximately µn by the strong law of large numbers. Thus, an additional effort is needed to justify the replacement of S n with µn. We start by proving an intermediate result.
Proof. Pick any increasing sequence (b n ) n∈N of positive numbers satisfying lim n→∞ b n = 1, for some c 1 , c 2 > 0 and for some n 0 ∈ N. One particular sequence satisfying these assumptions is given by b n = 1 − n −2 for n ∈ N (with c 1 = 2 and c 2 = 1/2 in (16)). Note that (16) entails Suppose we can prove that, for all ε > 0, Then, by the Borel-Cantelli lemma, for n large enough a.s. Since f is nonnegative and decreasing on [(1 − e −1 ) 1/2 , 1), we have, for all large enough n, Further, by the strong law of large numbers, for large n, the latter is estimated from above by Thus, noting that E|η k | ≤ 1, Using (16) and This proves that the first series on the right-hand side of (18) trivially converges, for it contains finitely many nonzero summands. By Markov's inequality and (16), the probability in the second series is upper bounded by , n → ∞.
In view of (17), this is the general term of a convergent series. Hence, the second series on the right-hand side of (18) converges. The proof of Lemma 5.2 is complete.
For b ∈ (0, 1) close to 1, δ as above and θ > 0, put Proof. Similarly to (7), summation by parts yields where, as in the proof of Theorem 1.1, T k = η 1 + . . . + η k for k ∈ N. By the strong law of large numbers, for b close to 1, One can check that Further, recall that, as ℓ → ∞, by the law of the iterated logarithm for standard random walks. Using this limit relation we infer According to (8), for b close to 1, With the help of (19) we obtain by an application of Lemma 3.2 with η = |ξ| and a = ∞. This in combination with (20) yields The proof of Lemma 5.3 is complete.
For b ∈ (0, 1) close to 1, put We claim that For the most part, this follows by repeating the proof of Lemma 5.3 with N 2 (b) replacing N 1, δ, θ (b), the only changes being that the second summand on the right-hand side of (21) and the right-hand side of (22) are O(((1 − b) δ log(1/(1 − b))) 1/2 ) as b → 1−. The last centered formula in combination with Lemma 5.2 enable us to conclude that This limit relation will be used in the proof of Proposition 5.7. Denote by F 0 the trivial σ-algebra and recall that, for k ∈ N, F k denotes the σ-algebra generated by (ξ j , η j ) 1≤j≤k and that, for k ∈ N 0 , we write E k (·) for E(·|F k ). and Proof. We only give a detailed proof of (24) and then explain which modifications are needed for a proof of (25). Proof of (24). For b ∈ (0, 1) and the same δ ∈ (0, 1) as before, put Plainly, for all ρ > 0, We first show that, for k ≥ 3, b close to 1 and ε > 0 to be defined below, Since lim n→∞ S n = +∞ a.s., we infer | inf i≥0 S i | < ∞ a.s. and thereupon Thus, given ε > 0 there exists a random variable b * such that b −S k−1 ≥ e −ε whenever 3 ≤ k ≤ N δ (b) and b ∈ (b * , 1) (of course, b −S k−1 ≥ 1 a.s. for all k ∈ N provided that ξ ≥ 0 a.s.). Thus, (26) does hold true in the present range of k. Let k ≥ N δ (b) + 1. By the strong law of large numbers, b S k−1 ≤ b δ(k−1) a.s. for b close to 1. Put We claim that the sequence (α k (b)) k≥N δ (b)+1 is nonincreasing. Indeed, We have used max b∈[0,1] (3b δ − b 3δ ) = 2 for the last step. Hence, for b close to 1, having utilized lim b→1− b δ(1−b 2δ ) −1 = e −1/2 for the last inequality. The proof of (26) is complete.
For b ∈ (0, 1), let K(b) be positive integers satisfying lim b→1− K(b) = ∞. In view of (26), for b close to 1, It is shown in the proof of Lemma 2.3 in [5 We treat the sums To deal with the second sum, we write, for b close to 1, having utilized the strong law of large numbers for the inequality. Analysis of I 1 . The limit relation lim b→1− b δN δ (b) = e −1/2 together with (28) and (30) in which we take K(b) = N δ (b) proves lim b→1− f (b)I 1 (b) = 0 a.s. Analysis of I 2 . Using (28) and (30) with Combining this with the first part of (19) we obtain lim b→1− f (b)I 2 (b) = 0 a.s. Analysis of I 3 . Write We have used Corollary 1.7.3 in [4] for the asymptotic equivalence. In view of (30) with s. The proof of (24) is complete.
Proof of (25). Similarly to (31), we obtain with the help of (a counterpart of (27)) and (29) that By the same reasoning, we also conclude that lim The proof of Lemma 5.4 is complete.
As usual, S c k (b) will denote the complement of S k (b), that is, Recall that 'i.o.' is a shorthand for 'infinitely often' and that, for a sequence of sets A 1 , A 2 , . . ., Then, for all ε > 0, The proof below follows the path of the proof of Lemma 3.6 in [22].
We start by showing that (recall that µ = 1 by convention). Indeed, By Theorem 1.1 with η = 1 a.s., which entails (32). For n ∈ N, put and define the event Equivalently, In view of (32), given β > 0, for large enough n. Thus, by the Borel-Cantelli lemma, Lemma 5.5 follows if we can check that n≥1 P(A n ) < ∞.
Proof. Throughout the proof we tacitly assume that the equalities and inequalities hold a.s. We start by writing, Summation by parts yields where T k = η 1 +. . .+η k for k ∈ N. For large enough n for which S N 1, δ, θ (bn)−1 ≥ δ(N 1, δ, θ (b n )−1) a.s. (this is secured by the strong law of large numbers) and, given ε > 0, having utilized (20) for the inequality. We are now passing to the analysis of I n,2 (b). By the strong law of large numbers, with the same δ ∈ (0, 1) there exists an a.s. finite τ such that max(δk, 1) ≤ S k ≤ (2 − δ)k for all k ≥ τ + 1.
We need some preparation to treat the remaining part of the sum. Using the fact that when is nonincreasing for b < 1 close to 1 we obtain on the event Combining this with a similar inequality on the event {ξ k < 0, τ ≤ k − 1} we arrive at for b and n as above. Thus, for b ∈ [b n , b n+1 ] and large n ∈ N, For all k ∈ N and all n ∈ N, where, as usual, x + = max(x, 0) and x − = max(−x, 0) for x ∈ R. For k ≥ τ + 1 and n ∈ N, by the mean value theorem for differentiable functions, and thereupon Thus, By Theorem 1.1, as n → ∞, Using (20) in combination with the property (a) of B for the first equality and the property (b) of B for the second we infer Invoking once again the property (b) of B we obtain lim n→∞ log(b n+1 /b n )N 1, δ, θ (b n ) = 0, whence lim n→∞ (b n+1 /b n ) (2−δ)N 1, δ, θ (bn) = 1. With this at hand we can argue as before to conclude that a.s.
Thus, we have proved that lim n→∞ I n,21 (b) = 0 a.s. Further, In view of (36), Invoking (38) and (20) in combination with lim n→∞ N 1, δ, θ (b n )(f (b n )) 2 = (1 + θ)(2δ) −1 we conclude that The last equality is justified as follows. Using subadditivity of x → x 1/2 on [0, ∞) we obtain, for large n, The property (a) of B entails and the first of these ensures is a consequence of the property (b) of B. Thus, the equality that we wanted to justify does indeed hold.
For the analysis of the second piece of I n,22 (b) we need an estimate similar to (37): for k, n ∈ N, This implies that Here, while the first equality is ensured by (20) and (38), the second is a consequence of the property (b) of B. The proof of lim n→∞ I n (b) = 0 a.s. is complete.
We proceed by analyzing J n (b): As before, appealing to the strong law of large numbers, we conclude that We have used (20) and (35) for the inequality and the property (a) of B and its consequences (39) for the equality. Invoking (8) we obtain, for large n and appropriate constant C > 0, Hence, an application of that lemma yields Using once again the property (a) of B and (39) in combination with the estimate for b The proof of Lemma 5.6 is complete.
We are ready to prove Proposition 5.1.
Proof of Proposition 5.1. We only prove (14), for (15) is a consequence of (14) with −η k replacing η k . By Lemmas 5.2 and 5.3 and (24), (14) is equivalent to The latter limit relation holds true by Lemma 5.5 in combination with (25) and the fact that , and Lemma 5.6.
Proposition 5.7. Under the assumptions of Theorem 1.5, and lim inf b→1− 1 − b 2 log log 1 Recall the notation: for b ∈ (0, 1) close to 1, Denote by B * the class of increasing sequences (b n ) n∈N of positive numbers satisfying the following properties: (a) lim n→∞ b n = 1 and lim n→∞ (1 − b n ) log n = 0; (b) for large n, N n+1 ≥ N 2 (b n ), where (c) for all a ∈ (0, 1) and some n 0 ∈ N, n≥n 0 log It was shown in Section 3 of [6] (see also pp. 180,181 and 184 in [5]) that the sequence (b n ) n≥3 given by b n := exp − 1 n! n j=2 (log j) 2 n k=3 log log k belongs to the class B * .
As in the proof of Proposition 5.1 we proceed via a sequence of lemmas. The latter is an immediate consequence of (1 − b 2 n )N n ∼ (log n) −1 → 0 as n → ∞ and lim sup n→∞ log log N n log log(1/(1 − b 2 n )) ≤ 1.
By the property (b) of B * , N n+1 ≥ N 2 (b n ) for large n which implies that, for large n, the random variables are independent. Hence, by the converse part of the Borel-Cantelli lemma, (45) is a consequence of We intend to prove (46). Fix any δ ∈ (0, 1). For each n ∈ N and t > 0, put q n (t) := 2t −2 log log For notational simplicity, we shall write q n for q n (t). Further, for each n ∈ N and each nonnegative integer k ≤ q n define numbers r k,n by r 0,n := 0, r k,n := inf j ≥ r k−1,n + 1 : b 2δr k−1,n n j−1 k=r k−1,n b 2k n ≥ σ 2 n q −1 n , k ∈ N, k ≤ q n − 1, where σ 2 n := N 2 (bn)−Nn−1 k=0 b 2k n , and r qn,n := N 2 (b n ) − N n + 1. One can check by a direct calculation that the numbers are well-defined and that, for k ∈ N, k ≤ q n , r k,n −r k−1,n −1 k=0 b 2k n ∼ r k,n − r k−1,n ∼ σ 2 n q n , n → ∞.
Recalling our convention that µ = s 2 = 1 it remains to prove that To this end, we first note that the random function b → k≥1 b S k−1 η k is a.s. continuous on [0, 1). Indeed, while the function b → b S k−1 η k is a.s. continuous on [0, 1), the latter series converges uniformly on [0, a] for each a ∈ (0, 1) with probability one. This follows from the inequality b S k−1 ≤ b δ(k−1) ≤ a δ(k−1) which holds for large k and b ∈ [0, a] and the fact that E k≥1 a δ(k−1) |η k | < ∞. Thus, the function b → f (b) k≥1 b S k−1 η k is a.s. continuous on ((1 − e −1 ) 1/2 , 1) with lim sup b→1− = 1 and lim inf b→1− = −1. This immediately entails (50) with the help of the intermediate value theorem for continuous functions.