Relative growth of the partial sums of certain random Fibonacci-like sequences

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1 Relative growth of the partial sums of certain random Fibonacci-like sequences Alexander Roitershtein Zirou Zhou January 9, 207; Revised August 8, 207 Abstract We consider certain Fibonacci-like sequences X n n 0 perturbed with a random noise. Our main result is that n X n X k converges in distribution, as n goes to infinity, to a random variable W with Pareto-like distribution tails. We show that log P W >x s = lim x log x is a monotonically decreasing characteristic of the input noise, and hence can serve as a measure of its strength in the model. Heuristically, the heavytailed limiting distribution, versus a light-tailed one with s = +, can be interpreted as an evidence supporting the idea that the noise is singular in the sense that it is big even in a slightly perturbed sequence. MSC200: Primary 60H25, 60J0, secondary 60K20. Keywords: random linear recursions, tail asymptotic, Lyapunov constant, Markov chains, regeneration structure. Introduction and statement of the main result Let η n n 0 be a sequence of independent Bernoulli random variables with P η n = = ε, P η n = 0 = ε for some ε 0,. We consider a sequence X n n 0 of real-valued random variables generated by the recursion with the initial conditions X 0 =, X = a, where X n+ = ax n + bη n X n, n N, a 0, and b > a 2 are given deterministic constants. The above construction is inspired by the models considered in [. The sequence X n can be thought as a perturbation with noise of its deterministic counterpart, which is defined through the recursion equation Z n+ = az n + bz n Dept. of Mathematics, Iowa State University, Ames, IA 500, USA; roiterst@iastate.edu Dept. of Mathematics, Iowa State University, Ames, IA 500, USA; zzhou@iastate.edu

2 and the initial conditions Z 0 =, Z = a. Throughout the paper we are interested in the dependence of model s characteristics on the parameter ε that varies while the recursion coefficients a, b are maintained fixed. It is not hard to check that lim n n Z n Z k = λ, where λ is a constant defined below in 8. In this paper we are primarily concerned with the asymptotic behavior of the following sequence W n := n X k, n N, 3 X n which describes the rate of growth of the partial sums relatively to the original sequence X n. Our main result is stated in the following theorem. Intuitively, it can be interpreted as a saying that while adding more noise to the input by increasing the value of ε yields more noise in the output sequence W n, the noise remains large for all, even arbitrarily small, values of the parameter ε > 0 in some rigorous sense. Theorem. Let W n be defined in 3. Then the following holds true: a There exists ε 0, such that i If ε 0, ε, then W n converges in distribution, as n goes to infinity, to a nondegenerate random variable W ε. ii If ε [ε,, then lim n P W n > x = for any x > 0, that is W ε = +. iii For any ε 0, ε, there exist reals s ε 0, and K ε 0, such that lim P W ε > xx sε = K ε. x b Furthermore, s ε is a continuous strictly decreasing function of ε on 0, ε, and lim s ε = while lim s ε = 0. 4 ε 0 ε ε The specific choice of the initial values X 0 = and X = a is technically convenient, but is not essential. In particular, while asserting it ultimately yields part a of Lemma 3, changing it wouldn t affect part b of the lemma. Theorem remains valid for an arbitrary pair X 0, X of positive numbers. See Remark 9 in Section 3 for details. To extend Theorem to a linear recursion under a more general than 2 assumption a 0, b > 0, one can consider X n = θ n X n with an arbitrary θ R such that aθ > 0 and 2 a < 2 θ < a + a 2 + 4b. The new sequence X n satisfies the recursion X n+ = ã X n + bη n Xn with ã = a/θ < and b = b/θ 2 > ã. Some other readily available extensions of Theorem are discussed in Section 5 below. The proof of Theorem is given in Section 3 below. Note that the theorem implies that the limiting distribution W ε has power tails as long as it is finite and non-degenerate. We remark that additional properties of the constants ε and s ε can be inferred from the auxiliary results discusses in Section 3 below. In particular, see Proposition 6 which provides 2

3 some information on the relation of s ε to the Lyapunov exponent and the moments of the reciprocal sequence Xn. For an integer n 0, let R n = X n X n+. 5 The sequence R n forms a Markov chain since is equivalent to R n = a + bη n R n. Notice that, since X 0 =, for n N we have Xn = n R n and W n+ = R n W n + R n or, equivalently, W n+ + = R n W n The proof of the assertion a-iii of Theorem is carried out by an adaption of the technique used in [ to obtain an extension of Kesten s theorem [6, 8 for linear recursions with i. i. d. coefficients to a setup with Markov-dependent coefficients. More specifically, to prove that the distribution of W ε is asymptotically power-tailed, we verify in Section 3 that Markov chain R n satisfies Assumption.5 in [. This allows us to borrow key auxiliary results from [, 3 and also use a variation of the underlying regeneration structure argument in [. See Lemma 7 in Section 3 below for details. The proof of Theorem relies in particular on the asymptotic analysis of the negative moments of X n more specifically, the function Λ ε t defined below in 8. First positive integer moments of X n can be in principle computed explicitly. We conclude this introduction with the statement of a result which is not directly connected to Theorem, but might be useful, for instance, for the statistical analysis of the sequence X n. Here and throughout this paper we use the notation E P to denote the expectation operator under the probability law P in order to distinguish it from the expectation E Q, where Q is introduced in Section 2 below. For ε [0,, let λ ε, = a + a 2 + 4b ε 2 > 0 and λ ε,2 = a a 2 + 4b ε 2 denote the roots of the characteristic equation λ 2 = aλ + b ε. We have: Proposition 2. For any integer n 0, a We have E P X n = λn+ ε, λ n+ ε,2. In particular, lim λ ε, λ n log E ε,2 n P X n = λ ε,. b We have lim n n log E P X n = aλ ε, + b ε. More precisely, E P X 2 n = c [aλ ε, + b ε n + c 2 [aλ ε,2 + b ε n 2 bn+ ε n 4b ε + a 2, < 0 7 where are the constants c and c 2 are chosen in a manner consistent with the initial conditions X 0 = and X = a. c Letting U n,k := E P X n X n+k for an integer k 0, U n,k = d n, λ k ε, + d n,2 λ k ε,2, 3

4 where are the constants d n, and d n,2 are chosen in a manner consistent with the initial conditions U n,0 = E P X 2 n and U n, = E P X n X n+ = a[ EP X 2 n+ b εe P X 2 n b n+ ε n. In particular, for any k N we have lim n n log E P X n X n+k = aλ ε, + b ε. The rest of the paper is organized as follows. Section 2 contains a preliminary discussion and an auxiliary monotonicity result with respect to the parameter ε about the Lyapunov constant of the sequence X n. The proof of Theorem is included in Section 3. The proof of Proposition 2 is deferred to Section 4. Finally, section 5 contains some concluding remarks regarding possible extensions of the results in Theorem. 2 Preliminaries: Lyapunov constant of X n This section includes a preliminary discussion which is focused on the random variable R n defined in 5 and the Lyapunov constant γε introduced below. The main purpose here is to obtain a monotonicity result in Proposition 4. The coupling construction employed to prove Proposition 4 is also used in Section 3, to carry out the proof of Propositions 6 and. Recall λ ε, and λ ε,2 from 7. In order to simplify the notation, denote λ := λ 0, = a + a 2 + 4b 2 and λ 2 := λ 0,2 = a a 2 + 4b. 8 2 Notice that the condition a + b > ensures λ >. Using the initial conditions Z 0 = and Z = a, one can verify that Z n = λn+ λ n+ 2, n 0. 9 λ λ 2 Using 9 one can obtain a Cassini-type identity Z n Z n+ Zn 2 = b n n+ see, for instance, Theorem 5.3 in [9 for the original Fibonacci sequence result and the identity Z n+ λ Z n = λ n+ 2. The alternating sign of the right-hand side in these two identities yields for k N, a Z 2k Z 2k 2 < Z 2k+ Z 2k < λ < Z 2k+2 Z 2k+ < Z 2k a2 + b Z 2k a. 0 Recall R n from 5. The unique stationary distribution for the countable, irreducible and aperiodic Markov chain R n can be obtained as follows. For n N, let Fix any k N. Then for a positive integer n > k we have P R n = Z k /Z k = P Tn = n k T n = sup{i n : η i = 0}. = P η n k = 0, η n k = η n k+ =... = η n 3 = η n 2 = = ε ε k. 4

5 Thus, lim n P R n = Z k /Z k = ε ε k. The stationary sequence R n n N can be extended into a double-infinite stationary sequence R n n Z [4. Let Q denote the law of the time-reversed stationary Markov chain R n n Z. We have established the following result: Lemma 3. Let S k = Z k Z k, k. Then: a For all n Z, we have P R n {S k : k N} =. b Furthermore, QR n = S k = ε ε k for any n Z, k N. Let γ = γε denote the Lyapunov exponent of the sequence X n, that is γ := lim n n log X n = lim n n E P log X n = E Q log, P a. s. and Q a. s. 2 R The existence of the limit along with the identities follow from results in [5. account that Z 0 =, 0 implies that Taking in γ = E Q log R = ε ε n log S n = n= The last formula can be compactly written as γ = ε E P log Z T, where ε 2 ε n log Z n. 3 T = + inf{k 0 : η k = 0} = inf{j : R j = /a}. 4 It follows from 3 and the fact that log S n is a bounded sequence, that γε is an analytic function of ε on [0,. In particular, lim γε = lim log S n = log λ and lim γε = log a. 5 ε 0 n ε We remark that the analyticity of γε on [0, follows directly from a general result in [2. For recent advances in numerical study of the Lyapunov exponent for random Fibonacci sequences see [0, 5 and references therein. We next prove formally the following intuitive result. Together with 5 it implies the existence of ε 0, such that γε > 0 if and only if ε < ε. Our interest to this phase transition steams from the result in Lemma 5 stated below in Section 3. Proposition 4. The function γε : [0, R is strictly decreasing. Proof. The proof is by a coupling argument. Fix any ε [0, and ε ε,. Let X n n 0 be the sequence introduced in, and define X n n 0 as follows: X 0 =, X = a, and where η n n= X n+ = ax n + bη n X n, n N, = min{η n, ξ n } and ξ n are i. i. d. random variables with the distribution { 0 with probability ε ε ξ n = ε with probability ε, ε 5

6 such that ξ n is independent of the σ-algebra σx 0, η 0, X, η,..., X n, η n, X n, X n+ for all n 0. Then P η n = 0 = ε, P η n = = ε, and hence the sequence X n n 0 is distributed according to the same law as X n n 0 with ε replacing ε in the definition of η n. To deduce that γε is a non-increasing function of ε, observe that by the coupling construction, η n η n for all n 0, and hence, by induction, X n X n for all n 0. To conclude the proof of the proposition it remains to show that γε is strictly decreasing. Toward this end, first observe that for any integer n 2 we have ζ n := η n η n = η n = 0, η n = = η n =, ξ n = 0, where A denotes the indicator function of the event A and the first equality serves as a definition of ζ n. Then, the following is an implication of Lemma 3 and 0 along with the fact which we have established that X n X n for all n 0 : [ log X n log X n = log X n 2 n axn + bx ζk n 2 log log X n [ n 2 axn + bx n 2 n 2 ζ k log ax n + b ax n ζk log [ n 2 + b ζk. 6 It follows then from 2 and the law of large numbers that with probability one, γε γε = lim n = E P ζ n log n The proof of the proposition is complete. log Xn log X n lim n n + b 3 Proof of the main result n 2 = ε ε log ζ n log + b + b > 0. The purpose of this section is to prove Theorem. The proof is divided into a sequence of lemmas. The critical exponent ε is identified and part a-i and a-ii of the theorem are proved in Proposition 6. The assertion in part a-iii of the theorem is verified in Lemma 7. Finally, the claim in part b is established in Proposition. Observe that under the stationary law Q the random variable W n = n n j=k R j has the same distribution as n k j=0 R j. Therefore one can write W ε = k j=0 R j. The following lemma is well-known, see for instance Theorem 2..2 especially display 2..6 and the subsequent Remark in [4. Lemma 5. For any ε 0, we have P W ε < = QW ε < {0, }. Moreover, P W ε < = if and only if γε = E Q log R > 0. 6

7 Let ε = sup{ε > 0 : γε > 0}. It follows from Proposition 4, the limits in 5, and the continuity of γε that ε 0, and γε = 0. 7 By virtue of Lemma 5, ε satisfies a-i and a-ii in the statement of Theorem. We proceed with the proof that a-iii of the theorem also holds true. Following [, 3, we are going to identify the critical exponent s ε in the statement of Theorem as the unique solution to the equation Λ ε s ε = 0, where for t 0 we define Λ ε t := lim n n log E Q R t... Rn. t It follows from Lemmas 2.6 and 2.8a in [ see especially display 2. in [ applied to the forward Markov chain R n n 0 that the above limit exists and in fact is not affected by the initial distribution of the Markov chain. In particular we have: Λ ε t = lim n n log E P X t 0 X t n = lim n n log E P Xn t The following proposition is a key ingredient in the proof of Theorem. Note that for any ε [0,, Λ ε 0 = 0 and, by virtue of the Cauchy-Schwarz inequality, Λ ε t is convex on [0,. In particular, the one-sided derivative Λ ε0 := lim t 0 Λ ε t/t is well-defined. Proposition 6. Let ε 0, be defined in 7. Then the following four statements are equivalent for ε 0, : i γε > 0, that is ε 0, ε. ii Λ ε0 < 0. iii There exists a unique s ε > 0 such that Λ ε s ε = 0. iv W ε is a P -a. s. finite and non-degenerate random variable. Proof. i ii If γε > 0, the ergodic theorem implies that for T defined in 4, Q-a. s., E Q T log R k = E Q T lim n n n log R k = E Q T E Q log R = ε γε < 0. k= Hence [6, 8 there exists a unique s > 0 such that E P X s T =. It can be shown see, for instance, display 2.43 in [ that this implies Λ ε s = 0 and hence Λ ε0 < 0. ii i Jensen s inequality implies that γ ε tλ ε t, and hence γε > 0 if Λ 0 < 0. ii iii For any t > 0, we have n E P Xn t E P X t η k ε n a tn, n k=2 and hence lim t Λ ε t = +. Since Λ ε t is a convex function with Λ ε 0 = 0, this proves the implication ii iii for an illustration, see Fig. below. i iv This is the content of Lemma 5. The proof of the proposition is complete. 7 8

8 Λ ε t ε ε 3 ε 2 Λ ε 0 = 0 ε Λ 0 t = t log λ s ε3 s ε2 s ε t Figure : Sketch of the graph of the convex function Λ ε t for 3 increasing parameter values ε < ε 2 < ε 3 within the range 0, ε and the extremal parameter values ε = 0, ε = ε. Using the notation introduced in the statement of Lemma 3, transition kernel of the timereversed Markov chain R n on the state space {S i : i N} can be written as follows: Hi, j := QR n = S j R n+ = S i = QR n+ = S i R n = S j QR n = S j QR n = S i ε ε j if i = = if i = j + 0 otherwise. Unfortunately, the infinite matrix Hi, j doesn t satisfy the conditions imposed in [, 3 or [3. More precisely, the kernel H doesn t satisfy the following strong Doeblin condition: H m i, j cµj for some m N, c > 0, a probability measure µ on N, and all i, j N. However, one can exploit the fact that transition kernel of the Markov chain R n does satisfy Doeblin s condition with m =, c = ε, and µ = δ, the degenerate distribution concentrated on j =. The proof of the following lemma is a mixture of arguments borrowed from [ and [3. The key technical ingredient of the proof is the observation that transition kernel of the forward Markov chain R n satisfies Assumption.2 in [3. Lemma 7. The claim in part a-iii of Theorem holds with ε introduced in 7. Proof. Let N 0 = 0 and then for i N, N i = sup{k < N i : R k = /a}. Note that the blocks R Ni+ +,..., R N i are independent and identically distributed for i 0. For i 0, let A i = R Ni + R Ni R R Ni N i R... R R Ni N i+ +2 N i+ + B i = R Ni R... R. Ni N i+ + 8

9 The pairs A i, B i, i 0, are independent and identically distributed under the law P. Moreover, it follows from 6 that W ε = A 0 + n= B i. A n n To prove Lemma 7 we will verify the conditions of the following Kesten s theorem for A i, B i i 0 under the law P. To enable a further reference see Section 5 below we quote this theorem in a more general setting with not necessarily strictly positive coefficients A n, B n than we actually need for the purpose of proving Lemma 7. Theorem 8. [6, 8 Let A i, B i i 0 be i.i.d. pairs of real-valued random variables such that i For some s > 0, E A 0 s = and E B 0 s log + B 0 <, where log + x := max{log x, 0}. ii P log B 0 = δ k for some k Z B 0 0 < for all δ > 0. Let W = A 0 + n= A n n i= B i. Then a lim t t s P W > t = K +, lim t t s P W < t = K for some K +, K 0. b If P B < 0 > 0, then K + = K. c K + + K > 0 if and only if P A 0 = B 0 c < for all c R. Recall T from 4. Observe that log B 0 = 0 k=n + log R k is distributed the same as T k= log R k = log Z log Z T +. Therefore the non-lattice condition ii of the above theorem holds in virtue of 9. Furthermore, since clearly P B 0 > > 0 and P B 0 < > 0, we have P A 0 = B 0 c < for all c R. It remains to verify condition i of the theorem. Recall S k introduced in the statement of Lemma 3. Let i=0 Hi, j := QR n+ = S j R n = S i = ε if j = ε if j = i + 0 otherwise be transition kernel of the stationary Markov chain R n. Between two successive regeneration times N i the forward chain R n evolves according to a sub-markov kernel Θ given by the equation That is, for i, j N, Hi, j = Θi, j + εj =. 9 Θi, j = QR = j, N > R 0 = i = { ε if j = i + 0 otherwise. Further, for any real s 0 define the kernels countable matrices H s i, j and Θ s i, j, i, j N, by setting H s i, j = Hi, jr t j and Θ s i, j = Θi, jr t j. For an infinite matrix A on N and a function f : N R let Af denote the real-valued function on N with Afi := j N Ai, jfj. Since the forward transition kernel H satisfies Assumption.2 in [3, it follows that from Proposition 2.4 in [3 that for all s 0 : 9

10 . There exist a real number α s > 0 and a bounded function f s : N R such that inf i N f s i > 0 and H s f = α s f s. 2. There exist a real number β s > 0 and a bounded function g s : N R such that inf i N g s i > 0 and Θ s f = β s f s. 3. β s 0, α s. Without loss of generality we can use the following normalization for the eigenfunctions: f s =. 20 Furthermore, it follows from Lemma 2.3 in [3 that α s and β s are spectral norms of infinite matrices H s and Θ s, respectively, and hence are uniquely defined. It follows from Proposition 6 see Lemma 2.3 in [3 that α sε =. In particular, since Λ ε is a continuous function of s, the spectral radius of Θ s regarded as an operator acting on the space of bounded function on N equipped with the sup-norm is strictly less than one on an interval 0, s ε for some s ε > s ε. Let I be the infinite unit matrix in N and h : N R be a function defined by hi = for all i N. For any s 0, s ε, and in particular for s = sε, we have: T E P B0 s = E P R k = n= k= T = E P R k = E P [ HT s, a s εrθ n s h = a s εi Θ s h. On the other hand, it follows from 9 that for any i N we have f sε i = H sε f sε i = Θ sε f sε i + εa sε f sε, and hence E P B sε 0 = f sε =, where for the second identity we used 20. Finally, adapting 2.45 in [ to our framework we obtain for any s s ε, s ε, [ n i s E P A s 0 = E P R j N = n = = = n= i= j=0 i s R j N = n [ n E P n= i= j=0 [ n i n s E P R j s N = n n= n= n= n s n s i= i= j=0 [ i n QR 0 = /a E Q R j s N = n, R 0 = /a j=0 n n n E P Rj s T = n = εa s n s Θ n i Θs i h <, i= j=n i 0 n= i=

11 where the last inequality is an implication of the fact that the spectral radius of the infinite positive matrix Θ s is strictly less than one. The proof of the lemma is complete. Remark 9. The initial conditions X 0 = and X = a guarantee that the measure P is Q conditioned on the event R 0 = /a. If R 0 has a different value, then A 0 and B 0 defined above are still independent of the i. i. d. sequence of pairs A n, B n n N, but the distributions of the pairs A 0, B 0 and A, B differ in general. Using a slightly more elaborated version of the arguments used in the proof of Lemma 7 cf. proof of Proposition 2.38 under assumption.6 in [ it can be shown that all the conclusions of Theorem remain valid for different strictly positive initial values X 0, X and that the only effect of changing initial conditions is on the value of the constant K ε. To conclude the proof of Theorem it remains to prove the claim in part b of the theorem. Using the representation of Λ ε t given in 8 and a variation of the coupling argument which we employed in order to prove Proposition 4, we first derive the following auxiliary result: Lemma 0. For any fixed t > 0, Λ ε t is a strictly increasing function of ε on [0,. Proof. Recall the notation introduced in the course of the proof of Proposition 4. It follows from the inequality in 6 that { n 2 + b ζk n 2 = exp ζ X n a 2 k log + b }. + b X n X n It follows then from Hölder s inequality that for any constants t > 0, p > and q > 0 such that + =, we have p q [ [E P E P X t n X n /p pt { n 2 E P exp qt Therefore, since ζ k are i. i. d. Bernoulli random variables, Λ ε t p Λ ε pt + lim sup n nq log E P { exp { exp n 2 qt ζ k log + b } /q. ζ k log + b = p Λ ε pt + q log E P qtζ log + b } = p Λ ε pt + { [ε q log ε exp qt log + b } = p Λ ε pt + q log [ε ε + b p Λ ε pt [ q ε ε + b qt + ε + ε qt, } + ε + ε where in the last step we used the inequality log x < x. Since Λ ε t is a continuous function of t, by letting p to approach one and thus q to approach infinity, we obtain that Λ ε t Λ ε t tε ε log + b > 0.

12 The proof of the lemma is complete. We now turn to the proof of part b of Theorem. Proposition. The critical exponent s ε is a strictly decreasing continuous function of ε on [0, ε. Furthermore, 4 holds true. Proof. The desired monotonicity of s ε follows directly from Lemma 0, see Fig. above. We will next show that s ε is a continuous function of ε on 0,. Due to the monotonicity of s ε, the following one-sided limits exist for any ε 0, ε : s + ε = lim δ ε s δ and s ε = lim δ ε s δ. The second limit, namely s ε, exists also for ε = ε. Set s ε := 0. If either s + ε > s ε or s ε < s ε for some ε [0, ε, then see Fig. above Λ δ t is not a continuous function of δ at any point t within the open interval s ε, s + ε or, respectively, s ε, s ε. To verify the continuity of s ε on 0, ε it therefore suffices to show that Λ ε t is a continuous function of ε for any fixed t > 0. We will use again the notation and the coupling construction introduced in the course of the proof of Proposition 4. Recall T n from, and let T n, n N, be the corresponding stopping times associated with the sequence X n. Let χ n = T n T n. The random variables χ n form a two-state Markov chain with transition kernel determined by and P χ n+ = χ n = 0 = P η n = η n = 0 = P ζ n+ = 0 = ε ε P χ n+ = χ n = = P η n = η n = 0 = ε. The stationary distribution π = π0, π of this Markov chain is given by π0 = ε ε and π = ε ε ε Similarly to 6, in virtue of Lemma 5 we have: X n n 2 X n a ξk = exp a X n { n 2 χ k log + b }. a 2 It follows then from Hölder s and Jensen s inequalities that for any constants t > 0, p > and q > 0 such that + =, we have p q E P t X n = [ /p { E P Xn pt [E P exp qt [ /p [ E P Xn pt [ E P X pt n { n 2 exp qt /p + b a 2 t 2 n 2 χ k log + b } /q a 2 E P χ k log + b } /q. a 2 n 2 E P χ k.

13 Since Markov chain χ n is aperiodic, its stationary distribution π is the limiting distribution. Thus, Λ ε t p Λ εpt + t log + b lim a P ξ 2 k = n = p Λ εpt + tπ log + b a 2 = p Λ εpt + t ε ε ε log + b. a 2 Since Λ ε t is a continuous function of t and p > is arbitrary, we conclude that 0 < Λ ε t Λ ε t t ε ε ε log + b, a 2 and thus, for a given t > 0, Λ ε t is a Lipschitz function of the parameter ε on any interval bounded away from zero. This completes the proof of the continuity of s ε on 0, ε. In particular, the second limit in 4 holds true. To complete the proof of the proposition it remains to prove that the first limit in 4 holds true, namely lim ε 0 s ε =. To this end it suffices to show that s ε > t for all ε > 0 small enough. To this end, observe that since 9 implies lim n S n = λ <, there exists k 0 N such that S k < 2 + λ < for all k > k0. For n N, let δ n = R n = S k with k > k 0 and let G n = σr, R 2,..., R n be the σ-algebra generated by the random variables R i with i n. Then, with probability one, we have for n 2, P δ n = 0 Gn P {η n k 2 = 0} G n 0 k k 0 k 0 P η n k 2 = 0 Gn = k0 + ε. 2 Denote u = 2 +λ and v = a. It follows from 8, 2, and 0 that for ε < +k 0 we have: Λ ε t = lim n n log E n Q i= R t i lim sup n log [ u t εk v t εk 0 +. n log E Q n i= u tσ i v t σ i It thus holds that Λ ε t < 0, and hence s ε > t, for all ε > 0 small enough. Since t > 0 is arbitrary, it follows that lim ε 0 s ε =. The proof of the proposition is complete. 4 Proof of Proposition 2 For k N, et F k = σx 0, η 0, X, η,..., X k 2, η k 2, X k, η k, X k, X k+ be the σ-algebra generated by the random variables η i with i k and X i with i k +. It follows from that η k is independent of F k. In the proof below, we will repeatedly use without further notice the fact E P Xη k = E P [X ε for a random variable X F k. Proof. 3

14 a In order to verify the claim, take the expectation on both sides of and recall 8, 9. b Take the square and then take the expectation on the both sides of, to obtain: E P X 2 n+ = a 2 E P X 2 n + b 2 ε 2 E P X 2 n + 2b εe P ax n X n = a 2 E P X 2 n + b 2 ε 2 E P X 2 n +2b εe P [ Xn+ bη n X n X n = a 2 E P X 2 n b 2 ε 2 E P X 2 n + 2b εe P X n+ X n = a 2 + 2b ε E P X 2 n b 2 ε 2 E P X 2 n +2b εe P h n, 22 where h n := X n X n+ Xn. 2 We will next derive a Cassini-type formula for E P h n. We have: [ ae P h n+ = E P axn X n+2 aep X 2 [ n+ = E P Xn+ bη n X n ax n+ + bη n X n axn+ 2 [ = E P bηn X n X n+ abη n X n X n+ b 2 η n η n X n X n Hence, = E P [ b εxn X n+ abη n X 2 n + h n b 2 εη n X n X n = E P [ b εxn X n+ ax n bη n X n abη n h n = abe P η n h n. E P X n X n+2 X 2 n+ = E P h n+ = be P η n h n =... = b n ε n E p η 0 h = n b n+ ε n. 23 Using the notation Y n = E P X 2 n and substituting 23 into 22, we obtain Y n+ = [ a 2 + 2b ε Y n b 2 ε 2 Y n + 2 b n+ ε n, from which the claim in b follows, taking in account that Y 0 = and Y = a 2. c For any k N, we have: U n,k+ = E P X n X n+k+ = ae P X n X n+k+ + be P η n+k X n X n+k = au n,k + b εu n,k. Furthermore, using notations introduced in the course of proving b, E P X n X n+ = a E b ε P X n X n+2 E P X 2 a n = [ EP h n+ + Yn+ 2 b εyn 2. a The proof of the proposition is complete. 4

15 5 Concluding remarks. We believe that K ε in the statement of Theorem is decreasing as a function of the parameter ε, but were unable to prove it. Some information about this constant can be derived from the formulas given in [2, 6 see also references in [2 using the recursion representation 6 of W n and the regeneration structure described in Section 3 see the proof of Lemma 7 there which reduces the Markov setup of this paper to an i. i. d. one considered in [2, 6, We think that s ε is a strictly convex function of ε on [0, ε, but were unable to prove it. Since lim ε 0 s ε = +, Fig. strongly suggests that the convexity holds for an interval of small enough values of ε within 0, ε. We believe that, with s ε set to zero, s ε is convex in fact on the whole interval 0, ε. 3. The linear model can serve as an ansatz in a general case. For instance, it seems plausible that a result similar to our Theorem holds for generalized Fibonacci sequences considered in [7. This is a work in progress by the authors. 4. Using appropriate variations of the above Proposition 6 and Lemma 7, Theorem can be extended to a class of recursions W n+ = θ m i=0 Rh i nl+i W n + Q n with arbitrary l, m N, positive reals h i, and suitable coefficients θ large enough by absolute value constant and Q n in general random. For instance, in the spirit of [9, one can consider sequences W n = n X2n 2 X 2k+X 2k+8 or W n = n Xn 2 k Xk 2. The former case corresponds to W n+ = Q n Wn + Q n with Q n = R 2n+ R 2n+2 R 2n+8 R 2n+9, and the later to W n+ = Q n Wn + with Q n = n Rn. 2 We leave details to the reader. References [ E. Ben-Naim and P. L. Krapivsky, Weak disorder in Fibonacci sequences, J. Phys. A , L30 L307. [2 D. Buraczewski, E. Damek, and J. Zienkiewicz, On the Kesten-Goldie constant, J. Difference Equ. Appl , [3 J. F. Collamore, Random recurrence equations and ruin in a Markov-dependent stochastic economic environment, Ann. Appl. Probab , [4 R. Durrett, Probability: Theory and Examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 200. [5 H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist , [6 C. M. Goldie, Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 99,

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