Functional limit theorems for divergent perpetuities in the contractive case

Let $\big(M_k, Q_k\big)_{k\in\mathbb{N}}$ be independent copies of an $\mathbb{R}^2$-valued random vector. It is known that if $Y_n:=Q_1+M_1Q_2+...+M_1\cdot...\cdot M_{n-1}Q_n$ converges a.s. to a random variable $Y$, then the law of $Y$ satisfies the stochastic fixed-point equation $Y \overset{d}{=} Q_1+M_1Y$, where $(Q_1, M_1)$ is independent of $Y$. In the present paper we consider the situation when $|Y_n|$ diverges to $\infty$ in probability because $|Q_1|$ takes large values with high probability, whereas the multiplicative random walk with steps $M_k$'s tends to zero a.s. Under a regular variation assumption we show that $\log |Y_n|$, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the $J_1$-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the $J_1$-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.


Introduction
Let M k , Q k k∈N be independent copies of a random vector M, Q with arbitrary dependence of the components, and let X 0 be a random variable which is independent of M k , Q k k∈N . Then the sequence X n n∈N 0 defined by In the case that X 0 = 0 a.s. it is easily seen that X n has the same law as Y n for each fixed n.
It is also well-known what happens in the 'trivial cases' when at least one of conditions (1.2) and (1.3) does not hold. (a) If P{M = 0} > 0, then τ := inf{k ∈ N : M k = 0} < ∞ a.s., and the perpetuity trivially converges, the limit being an a.s. finite random variable τ k=1 Π k−1 Q k . Plainly, its law is a unique invariant measure for (X n ). (b) If P{Q = 0} = 1, then k≥1 Π k−1 Q k = 0 a.s. (c) If P{Q + M r = r} = 1 for some r ∈ R, then either δ r is a unique invariant probability measure for (X n ) or every probability law is an invariant measure, or every symmetric around r probability law is an invariant measure (see Theorem 3.1 in [8] for the details).
Under assumptions (1.2), (1.3) and (1.4) the Markov chain X n has a unique invariant probability measure which is the law of the perpetuity. Equivalently, the law of Y is a unique solution to the stochastic fixed-point equation where the vector (M, Q) is assumed independent of Y , sometimes called the random difference equation. Equations (1.5) appear in diverse areas of both applied and pure mathematics and various properties of Y have attracted considerable attention.
Papers [1,8,18] give pointers to relevant literature. For (X n ) defined by (1.1) we write X v n to indicate that X 0 = v for v ∈ R. If the first part of (1.4) is in force we infer |X v n − X w n | = Π n |v − w| → 0 a.s. as n → ∞, for any v, w ∈ R. Therefore, the case when lim n→∞ Π n = 0 a.s. will be called contractive.
In the present paper we are interested in the case when conditions (1.2), (1.3) and lim n→∞ Π n = 0 a.s. and I = ∞ (1.6) hold, i.e., the model is still contracting, yet the second condition in (1.4) is violated. By Theorem 2.1 in [8] (Y n ) is then a divergent perpetuity in the sense that The purpose of the present paper is to prove functional limit theorems for the Markov chains (X n ) and for the divergent perpetuities (Y n ) under the aforementioned assumptions.
As far as we know Grincevičius [9] was the first to prove a limit theorem for Y n in the case E log |M | = 0 under the assumption that M > 0 a.s. Also, weak convergence of one-dimensional distributions of divergent perpetuities has been investigated in [3,10,13,15] under various assumptions on M and Q. To the best of our knowledge, (a) functional limit theorems for divergent perpetuities have not been obtained so far; (b) [13] is the only contribution to case (1.6) which deals with one-dimensional convergence. We would like to stress that outside the area of limit theorems we are only aware of two papers [12] and [19] which investigate case (1.6). Unlike (1.6) the critical non-contractive case E log |M | = 0 has received more attention in the literature, see [2,4,5,6,9,10,15].
Assuming that the tail of log − |M | is lighter than that of log + |Q| we state two functional limit theorems thereby covering a variety of situations. In particular, we do not require finiteness of E log |M |. Under (1.6) the complementary case is also possible where the tail of log − |M | is not lighter than that of log + |Q|. Take, for instance, P{log − |M | > x} ∼ x −α log x, x → ∞, and P{log |Q| ∈ dx} = αx −α−1 ½ (1,∞) dx for some α ∈ (0, 1). Even though this situation is beyond the scope of the present work we note without going into details that it is unlikely that there is functional convergence in the Skorokhod space equipped with one of the standard topologies like J 1 or M 1 . Also, it is worth to stress that unlike some previous papers on limit theorems for perpetuities we allow M and Q to take values of both signs.
For c > 0 and α > 0, let Let D := D[0, ∞) denote the Skorokhod space of right-continuous functions defined on [0, ∞) with finite limits from the left at positive points. Throughout the paper we use '⇒' to denote weak convergence in the Skorokhod space D equipped with the J 1 -topology. We write '⇒ in S' to denote weak convergence in a space S other than D. Also, we stipulate hereafter that the supremum over the empty set is equal to zero. Theorem 1.1 treats the situation in which both M k 's and Q k 's affect the limit behavior of the processes in question, whereas in the situation of Theorem 1.5 only the contribution of Q k 's persists in the limit. where g(t) := −t, t ≥ 0.
Remark 1.2. Conditions (1.9) and (1.11) ensure that the paths of log Y [n·]+1 | and log X [n·]+1 | belong to D. While a simple sufficient condition for (1.9) to hold is continuity of the law of Q, (1.11) holds if either X 0 = 0 a.s. and the law of Q is continuous or the law of X 0 is continuous. Condition (1.9) ((1.11)) is not needed if (a) we replace log with log + in (1.10) ((1.12)); (b) consider weak convergence in D(0, ∞) rather than D. The same remark also concerns Theorem 1.5 given below.
Remark 1.3. Since X n d = Y n for each n ∈ N provided that X 0 = 0 a.s., the onedimensional distributions of the limit processes in (1.10) and (1.12) must coincide. Moreover, they can be explicitly computed and are given by Indeed, for x ≥ 0, the probability on the left-hand side equals P N (c/a,1) (t, y) : t ≤ u, −t+y > x = 0 = exp −EN (c/a,1) (t, y) : t ≤ u, −t+y > x because N (c/a,1) (t, y) : t ≤ u, −t + y > x is a Poisson random variable. It remains to note that Remark 1.4. Theorem 5(ii) in [13] states that, for fixed a > 0, By an Abelian-Tauberian argument the last relation is equivalent to (1.8). This implies that convergence (1.14) follows from (1.10) and (1.13).
Theorem 1.5. Suppose that P{M = 0} = 0, lim n→∞ Π n = 0 a.s., and that for some α ∈ (0, 1] and some ℓ slowly varying at ∞. Let (b n ) be a positive sequence which satisfy lim n→∞ nP{log |Q| > b n } = 1. In the case α = 1 assume additionally 1 and if condition (1.11) holds, then Remark 1.6. Theorem 5(iii) in [13] states that, for fixed a > 0, Therefore, (1.19) follows from (1.17) after noting that The rest of the paper is structured as follows. In Section 2 we state and prove Theorem 2.1, a deterministic result which is our key tool for dealing with the functional limit theorems. With this at hand, Theorem 1.1 and Theorem 1.5 are then proved in Section 3 and Section 4, respectively.

Main technical tool
for all δ > 0 and all T > 0. The M p is endowed with the vague topology. Denote by M * p the set of ν ∈ M p which satisfy where the signs + and − are arbitrarily arranged, and (c n ) is some sequence of positive numbers. The definition of F n in the case of empty sum stems from the fact that we define (A3) if not all the signs under the sum defining F n are the same, then and sup for each T > 0 such that ν 0 ({T }, (0, ∞]) = 0 and small enough γ > 0; Then lim in D in the J 1 -topology.
Proof. It suffices to prove convergence (2.5) in D[0, T ] for any T > 0 such that If all the signs under the sum defining F n are the same, then for all t ∈ [0, T ]. In this case, (2.5) is a trivial consequence of Theorem 1.3 in [11] which treats the convergence lim In what follows we thus assume that not all the signs are the same.
Pick now γ > 0 so small that and that sup τ (0) With this at hand we have where d T is the standard Skorokhod metric on D[0, T ]. We treat the terms on the right-hand side of (2.8) separately. 1st term. The relation lim n→∞ sup t∈[0, T ] |λ n (t) − t| = 0 is easily checked.
2nd term. We denote the second term by I n (γ) and use inequality having utilized log(1 + x) ≤ log x + 1/x, x > 0 and that λ n (τ ≤ T . The first term on the right-hand side of (2.9) converges to zero in view of (2.4). As to the second, we apply Theorem 1.3 in [11] as n → ∞. The latter goes to zero as γ → 0 because f 0 = 0 by assumption. Finally, the last term on the right-hand side of (2.9) tends to zero as n → ∞ for the principal factor of exponential growth does so as a consequence of (2.10) and the assumption sup τ (0) Summarizing we have proved that lim   With these at hand we can proceed as follows In view of (2.7) and (2.11) the right-hand side tends to zero uniformly in t ∈ [0, T ] as n → ∞. We already know that Recalling that 4th term. In the proof of Theorem 1.3 in [11] it is shown that 4 of continuity of f 0 . Of course, the right-hand side of the last inequality tends to zero on sending |ρ| and γ to zero.
Collecting pieces together and letting in (2.8) n → ∞ and then |ρ| and γ tend to zero we arrive at the desired conclusion lim n→∞ d T (F n (f n , ν n ), G(f 0 , ν 0 )) = 0.

Proof of Theorem 1.1
Proof of (1.10). We first show that where h(t) = 0, t ≥ 0. To this end, we intend to check that conditions (1.  7) we must have E log + |Q| < ∞. This contradiction completes the proof of (3.1).
Next we check (2.2). Our argument is similar to that given on p. 223 in [17]. We fix any T > 0, δ > 0 and use the representation where (U i ) are i.i.d. with the uniform [0, T ] distribution, (V j ) are iid with P{V 1 ≤ x} = (1−δ/x) ½ (δ,∞) (x), and N has the Poisson distribution with parameter T c/(aδ), all the random variables being independent. It suffices to prove that This is a consequence of the fact that −U 1 + V 1 has a continuous distribution which An analogous working leads to the conclusion that N (c/a,1) does not have clustered jumps a.s., i.e., (A2) holds. The last thing that needs to be checked is condition (2.3). Arguing as in Remark 1.3 we infer for any T > 0 and any γ ∈ (0, T ). Proof of (1.12). Without loss of generality we assume that X 0 = 0 a.s. and use the representation where Π * k := Π −1 k , k ∈ N 0 and Q * k := Q k /M k (with generic copy Q * ), k ∈ N. where g(t) = −t, t ≥ 0. Further, write, for ε ∈ (0, 1) and x > 0, Multiplying the inequality by x, sending x → ∞ and then ε → 0 yields are easily checked. Also, we have lim n→∞ Π * n = ∞ a.s. Hence | n k=1 Π * k−1 Q * k | P → ∞ as n → ∞ by Theorem 2.1 in [8]. Arguing in the same way as in the proof of (1.10) we see that An application of Theorem 2.1 gives 5 Now (1.12) follows by a combination of the last two relations and (3.5).

Proof of Theorem 1.5
The proof proceeds along the lines of that of Theorem 1.1 but is simpler for the contribution of M k 's is negligible. Therefore we only provide details for fragments which differ principally from the corresponding ones in the proof of Theorem 1.1.
Observe that lim Indeed, since (b n ) is a regularly varying sequence of index 1/α, this is trivial when α ∈ (0, 1). If α = 1, this follows from the relation b n /n ∼ ℓ(b n ) as n → ∞ and our assumption that lim x→∞ ℓ(x) = ∞.
Proof of (1.17). As far as is concerned which is the counterpart of (3.1) we have to check two things that are not obvious in the case when E log − |M | = ∞: condition (1.3) and Assume first that P{Q+M r = r} = 1 for some r = 0. In view of |Q−r| = |M ||r|, the tails of log + |Q| and log + |M | must exhibit the same asymptotics. However, this is not a case, for the tail of log + |Q| is heavier than that of log + |M |.
Next, according to (1.16), for any B > 0 there exists x 0 > 0 such that Thus, (4.2) holds. 5 We omit details which are very similar to but simpler than those appearing in the proof of (1.10).
To proceed we recall the already used notation S k := log |Π k | and η k+1 := log |Q k+1 |, k ∈ N 0 . According to Corollary 4.19 (ii) in [16] condition (1.15) entails in M p . If we can prove that where h(t) = 0, t ≥ 0, then relations (4.3) and (4.4) can be combined into the joint convergence By the Skorokhod representation theorem there are versions which converge a.s. Retaining the original notation for these versions we apply Proposition , c n = b n and the signs ± defined by sgn(Π k Q k+1 ) which gives (1.17) with log replaced with log + . The latter in combination with (4.2) proves (1.17).
It only remains to check (4.4). To this end, it suffices to prove that (4.5) follows if we prove that lim n→∞ (S ± n /b n ) = 0 in probability. While doing so, we treat two cases separately. Case when E log − |M | < ∞. Then necessarily E log + |M | < ∞ for otherwise lim n→∞ Π n = ∞ a.s. Therefore we have lim n→∞ n −1 S ± n = E log ± |M | by the strong law of large numbers. Invoking (4.1) proves (4.5).
Case when E log − |M | = ∞. Condition (1.16) entails lim n→∞ n bn E (log − |M |) ∧ b n = 0. Since   Left with proving that lim n→∞ (S + n /b n ) = 0 in probability we suppose immediately that E log + |M | = ∞ for the complementary case can be treated in exactly the same way as above (use the strong law of large numbers). Since lim n→∞ S n = −∞ a.s. by the assumption, Lemma 8.1 in [14] tells us that lim n→∞ S + n /S − n = 0 a.s. which together with lim n→∞ (S − n /b n ) = 0 in probability implies lim n→∞ (S + n /b n ) = 0 in probability. The proof of (4.4) is complete. Hence so is that of (1.17). Proof of (1.18) follows the pattern of that of (1.12) but is simpler. Referring to (1.12) the only things that need to be checked are that where h(t) = 0, t ≥ 0, and that P{log |Q| − log |M | > x} ∼ P{log |Q| > x} ∼ x −α ℓ(x), x → ∞.
To prove the first of these, write Since E log − |M | < ∞ entails E log + |M | < ∞, the same argument proves (4.8) for the tail of log + |M |. Case E log − |M | = ∞ and E log + |M | < ∞. It suffices to check (4.8) which is a consequence (1.16).