Kesten s power law for difference equations with coefficients induced by chains of infinite order

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1 Kesten s power law for difference equations with coefficients induced by chains of infinite order Arka P. Ghosh 1,2, Diana Hay 1, Vivek Hirpara 1,3, Reza Rastegar 1, Alexander Roitershtein 1,, Ashley Schulteis 4, Jiyeon Suh 5 Abstract We consider the equation R n = Q n + M n R n 1, with random coefficients (Q n, M n ) n Z induced by chains of infinite order. We study the asymptotic behavior of the distribution tails of the stationary solution R to this equation and prove a power law for P (R > t). Keywords: random equations, power tails, chains of infinite order, chains with complete connections, regenerative structure, Markov representation MSC: 60K15, 60J Introduction and statement of results Let (Q n, M n ) n Z be a stationary and ergodic sequence of pairs of real-valued random variables, and consider the recurrence relation R n = Q n + M n R n 1, n N, R n R. (1) This equation has a wide variety of real world and theoretical applications, see for instance (Embrechts and Goldie, 1993; Vervaat, 1979). If E (log M 0 ) < 0 and E ( log + Q 0 ) < (where we denote x + := max{x, 0} for x R) then for any R 0, the limit law of R n is that of R = Q 0 + n=1 Q n 1 n i=0 M i, and it is the unique initial distribution under which (R n ) n 0 is stationary (Brandt, 1986). The distribution tails of R are shown to be power tailed in (Kesten, 1973; Goldie, 1991), provided that the pairs (Q n, M n ) n Z form an i.i.d. sequence. Extensions of this result to Markovian coefficients have been obtained in (de Saporta, 2005; Roitershtein, 2007). The goal of this paper is to consider a non-markov setup which is desirable in This work was partially done during Summer 09 Research Experience for Undergraduate (REU) Program at Iowa State University. D.H, R.R, and A.S. were partially supported by NSF Award DMS J.S. thanks the Department of Mathematics at Iowa State University for the hospitality and support during her visit there. Corresponding author. roiterst@iastate.edu 1 Department of Mathematics, Iowa State University, Ames, IA 50011, USA 2 Department of Statistics, Iowa State University, Ames, IA 50011, USA 3 Department of Economics, Iowa State University, Ames, IA 50011, USA 4 Department of Math, Physics and CS, Wartburg College, Waverly, IA 50677, USA 5 Department of Math, Grand Valley State University, Allendale, MI 49401, USA Preprint submitted to Statistics & Probability Letters April 2, 2010

2 many, especially financial, applications. See for instance the concluding discussion in (Perrakis and Henin, 1974). Definition 1.1. (Lalley, 1986) A C-chain is a stationary process (X n ) n Z taking values in a finite set (alphabet) D such that (i) For any i 1, i 2,..., i n D, P (X 1 = i 1, X 2 = i 2,..., X n = i n ) > 0. (ii) For any i 0 D and any sequence (i n ) n 1 D N, the following limit exists: lim P (X 0 = i 0 X k = i k, 1 k n) = P (X 0 = i 0 X k = i k, k 1), n where the right-hand side is a regular version of the conditional probabilities. (iii) (fading memory) For n 0 let { } P (X 0 = i 0 X k = i k, k 1) γ n := sup P (X 0 = j 0 X k = j k, k 1) 1 : i k = j k, k = 1,..., n. Then, the numbers γ n are all finite and lim sup n log γ n /n < 0. C-chains are a particular case of chains of infinite order (chains with complete connections). For general accounts on these processes we refer to (Iosifesku and Grigoresku, 1990; Kaijser, 1981). The distributions of C-chains (a particular case of g-measures introduced in (Keane, 1972)) are Gibbs states in the sense of Bowen (also known as Dobrushin-Lanford-Ruelle states), see e.g. (Bowen, 1975; Lalley, 1986). We have, see also (Berbee, 1987; Fernández and Maillard, 2004), Theorem A. (Lalley, 1986) Let (X n ) n Z be a C-chain with alphabet D. Let S = n 1 Dn and ζ : S D be the projection into the last coordinate, i.e. ζ ( (s 1,..., s n ) ) = s n. Then there exist a stationary irreducible Markov chain (Y n ) n Z defined on the state space S and constants r N, δ > 0 such that: (i) The following Markov representation holds: X n = ζ(y n ), n Z. Furthermore, for any (y n ) n N D N and s N, (ii) P ( Y n+1 = (x 1, x 2,..., x t ) Y n = (y 1, y 2,..., y s ) ) = 0 unless either t = 1 or t = s + 1 and x i = y i for all i s, (iii) P ( Y n+1 = (y) Y n+1 D, Y n = (y 1, y 2,..., y s ) ) = P ( Y 0 = (y) Y 0 D). (iv) P ( Y n+1 D Y n = (y 1, y 2,..., y mr ) ) = δ, for all m N. Moreover, without loss of generality, we can assume that (v) r is even (see the beginning of the proof in (Lalley, 1986, p. 1266)). Let (X n ) n Z be a C-chain and denote X = (X n ) n 0. Our main result is: Theorem 1.2. Let (Q n,i, M n,i ) n Z,i D R 2 be a sequence independent of (X n ) n Z such that for any j D, (Q n,j, M n,j ) n Z are i.i.d., and Q n = j D Q n,j I {Xn =j} = Q n,xn and M n = j D M n,j I {Xn =j} = M n,xn. (2) Assume that: 2

3 (i) P ( Q 0 < q 0 ) = 1 and P (m 1 ) 0 < M 0 < m 0 = 1 for some constants q0 > 0, m 0 > 0. ( (ii) lim sup 1 n log E n 1 ) ( i=0 M i β1 0 and lim sup 1 n log E n 1 ) i=0 M i β2 < 0 for some n n constants β 1 > 0 and β 2 > 0. (iii) P (log M 0 = δ k for some k Z M 0 0) < 1 for all δ > 0. Then, with parameter α > 0 defined below in (4), the following hold: (a) lim t α P (R η > t X ) = K η (X ) for bounded K η : D Z+ R +, η = 1, 1. Moreover, t the convergence is uniform on X. (b) If P (M 0 < 0) > 0 then P ( K 1 (X ) = K 1 (X ) ) = 1. (c) If P (Q 0 > 0, M 0 > 0) = 1 then P ( K 1 (X ) > 0 ) = 1. (d) For η { 1, 1}, if P ( K η (X ) > 0 ) > 0 then P ( K η (X ) > 0 ) = 1. (e) P ( K 1 (X ) = K 1 (X ) = 0 ) = 1 iff there is a function Γ : D R s. t. P ( Q 0 + Γ(X 0 )M 0 = Γ(X 1 ) ) = 1. (3) Moreover, if P ( Q 0,i = q, M 0,i = m ) < 1 for all pairs (q, m) R 2 and i D, then P ( K 1 (X ) = K 1 (X ) = 0 ) = 1 iff P ( Q 0 + cm 0 = c ) = 1 for a constant c R. It turns out (see below, in Section 2) that both lim sup s in (iii) are in fact limits, and the parameter α > 0 is (uniquely) determined by Λ(α) = 0, 1 ( where Λ(β) = lim n n log E n 1 Π i=0 M i β). (4) We remark that condition (3) is a natural generalization of the criterion that appears in the i.i.d. case, see Theorem B below. Note that (2) defines a Hidden Markov Model (HMM) when X n is a Markov chain. In fact, (Q n, M n, Y n ) n Z can be thought as a HMM associated with Y n by X n = η(y n ). The proofs in (Kesten, 1973; Goldie, 1991) and (Roitershtein, 2007; de Saporta, 2005) rely on renewal theory for Markov additive process (Markov random walk) V n = n 1 i=0 log M i under exponential tilt. Following the idea of (Mayer-Wolf et al., 2004), Theorem 1.2 is obtained in Section 2 rather directly from its i.i.d. prototype (cited as Theorem B below) by applying a Doeblin s cyclic trick and associating R to a linear recursion with i.i.d. coefficients. Though a reduction to the main results of (Roitershtein, 2007) is possible, it would anyway entail considerable extra work. The approach taken here enjoys the finite range and mixing properties of X n, in addition to the existence of Y n. It is much lighter than those used in (Roitershtein, 2007) for general Markov chains and in (de Saporta, 2005) for a finite state case, both built on techniques of (Goldie, 1991). 2. Proof of Theorem 1.2 Consider the backward chain Z n = Y n, n Z, with transition matrix H given by H(x, y) = P (Y n = y Y n+1 = x) = P (Y0=y) P (Y 0 =x) P (Y n+1 = y Y n = x). Fix y S and a constant r (0, 1). Let (η n ) n Z be a sequence of i.i.d. coins independent of both 3

4 (Z n ) n Z and ( Q n,i, M n,i )n Z,i D, s. t. P (η 0 = 1) = r, P (η 0 = 0) = 1 r. Let N 0 = 0 and N i = inf{n > N i 1 : Z n = y, η n = 1}, i N. The blocks (Z Ni,..., Z Ni+1 1 ) are independent for i 0 and identically distributed for i 1. Between two successive regeneration times N i the chain (Z n ) n 0 evolves according to a sub-markov kernel Θ given by H(x, y) = Θ(x, y) + r1 {y=y }H(x, y), i.e. Θ(x, y) = P (Z 1 = y, N 1 > 1 Z 0 = x). (5) Theorem A implies that E ( ) e βn 1 Z0 is uniformly bounded for some β > 0 (see the paragraph following Theorem 1 in (Lalley, 1986)). For i 0, let A i = Q Ni + Q Ni +1M Ni Q Ni+1 1M Ni M Ni M Ni+1 2 B i = M Ni M Ni M Ni+1 1. The pairs (A i, B i ) are independent for i 0, identically distributed for i 1 and R = A 0 + n=1 A n 1 n i=0 B i. To prove Theorem 1.2 we will verify the conditions of the following theorem for (A i, B i ) i 1. Theorem B. (Kesten, 1973; Goldie, 1991) Let (A i, B i ) i 1 be i.i.d. and (i) For some α > 0, E ( A 1 α ) = 1 and E ( B 1 α log + B 1 ) <. (ii) P (log B 1 = δ k for some k Z B 1 0) < 1 for all δ > 0. Let R = A 1 + n=2 A n 1 n i=1 B i. Then (a) lim t α P ( ) R > t = K+, lim t α P ( ) R < t = K for some K +, K 0. t t (b) If P (B 1 < 0) > 0, then K + = K. (c) K + + K > 0 if and only if P ( A 1 = (1 B 1 )c ) < 1 for all c R. Let H β (x, y) = H(x, y)e( M 0,ζ(y) β ) and Θ β (x, y) = Θ(x, y)e( M 0,ζ(y) β ). Notice that E ( n i=0 M i β 1 {Zn =y} Z 0 = x ) = E ( M 0 β Z 0 = x ) Hβ n (x, y) and E ( 1 {n<n1 } n M ) i β 1 {Zn =y} Z 0 = x = E ( M 0 β Z 0 = x ) Θ n β(x, y), y y. i=0 (Roitershtein, 2007, Lemma 2.3 and Proposition 2.4) show that (4) holds and that the spectral radius of H α is 1 while the spectral radius of Θ α is less than 1. Hence, see the proof of (2.43) in (Mayer-Wolf et al., 2004), we have: Lemma 2.1. E( B 1 α ) = 1. We next show that log B 0 is a non-lattice random variable if y D. Lemma 2.2. Assume y D 1 and let V = N 1 1 n=0 log M n = log B 0. Then for any δ > 0 we have P ( V δ Z Z 0 = y ) < 1, where δ Z := {k δ : k Z}. Proof. For y D and k N, let y (k) denote (y 1,..., y k ) D k such that y 2 = = y k = y and y 1 = y. By Lalley s Theorem A, for any y D, P (Z 0 = y (1), Z 1 = y (m),..., Z m 1 = Z N1 1 = y (2) ) > 0 (6) 4

5 for some m > 1. Therefore, P ( V δ Z Z 0 = (y ) ) = 1 along with (6) imply: (i) using first y = y, P ( m log M 0,y δ Z Z 0 = (y ) ) = 1; (ii) using then a general y D, P ( (m 1) log M 0,y + log M 0,y δ Z Z0 = (y ) ) = 1. Therefore, we would have P ( m(m 1) log M 0,y δ Z Z 0 = (y ) ) = 1 for any y D, contradicting condition (iv) of Theorem 1.2. The proof of the next lemma is verbatim the proof of (2.44) in (Mayer-Wolf et al., 2004), and therefore is omitted. Lemma 2.3. β > α s. t. E [( N 1 1 n 1 n=0 i=0 M i ) β ] Z 0 is bounded. Completion of the proof of Theorem 1.2. (a) (d) It follows from Lemmas that the conclusions of Theorem B can be applied to (A 1, B 1 ). Claims (a) through (d) of Theorem 1.2 follow then from the identity R = A 0 + B 0 R and the independence of (A0, B 0 ) of R under the conditional measures P ( Z 0 = x). In particular, for η { 1, 1}, lim P ( R η > t Z 0 ) = E ( B 0 α (1 t {B0 η>0}k {B0 η<0}k ) ) Z 0 (7) Notice that Theorem A-(v) implies P (N 1 is odd) > 0 and P (N 1 is even) > 0. Therefore, P (M 0 < 0) > 0 yields P (B 0 < 0 Z 0 = x) > 0 for all x S. (e) First, (3) implies P ( R n = Γ(X n ) ) = 1 provided that R 0 = Γ(X 0 ). Hence if (3) is true and R 0 = Γ(X 0 ), then R n can take only a finite number of values from {Γ(i) : i D} and cannot converge in distribution to a power-tailed random variable. Therefore, (3) implies P ( K 1 (X ) + K 1 (X ) = 0 ) = 1. Assume now that P ( K 1 (X )+K 1 (X ) = 0 ) = 1. Then (7) along with Theorem 1.2-(b) imply that K + = K = 0. By Theorem B, K + = K = 0 iff P ( A 1 + c(y )B 1 = c(y ) ) for a constant c(y ) R, in which case 1 = P ( R = c(y ) ) = P ( R = c(y ) Z 0 = y ) = P ( Q 0 + M 0 R = c(y ) Z 0 = y ) with R = R Q 0 M 0. Since R is conditionally independent of (Q 0, M 0 ) given Z 0, it follows that for some constant c 1 (y ) R, P ( R = c1 (y ) Z 0 = y ) = P ( R = c 1 (y ) Z 1 = y ) = 1 (8) P ( Q 0,ζ(y ) + M 0,ζ(y )c 1 (y ) = c(y ) ) = 1. (9) It follows from (8) and the Markov property that P ( R = c 1 (y ) Z 0 = y ) = 1 for any y S such that P (Z 0 = y Z 1 = y ) > 0, and hence P ( R = c(y ) Z 1 = x ) = 1 x S s. t. P (Z 0 = y Z 1 = x) > 0. (10) Thus P ( R = c 1 (x) = c(y ) Z 1 = x ) = 1 whence P (Z 0 = y Z 1 = x) > 0. Since y S is arbitrary, P ( Q 0,ζ(Z0 ) + M 0,ζ(Z0 )c 1 (Z 0 ) = c 1 (Z 1 ) ) = 1 due to (9). This is exactly (3) once one sets Γ(Y n ) = c 1 (Z n ). Suppose now in addition that (Q 0,i, M 0,i ) is non-degenerate for any i D. Since the system of equations q + mc 1 = c, q + m c 1 = c for unknown variables q and m has a unique solution unless c 1 = c 1, it follows from (9) that c 1 depends on ζ(y ) only. Let Γ(i) = c 1 (y ) for i = ζ(y ). Then, (9) and (10) imply P ( Q 0 + M 0 Γ(X 0 ) = Γ(X 1 ) ) = 1. Hence, in virtue of (i) of Definition 1.1, P (Γ(X n ) = c) = P ( Q 0 + M 0 c = c ) = 1 for some constant c R. 5

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