On the almost sure convergence of series of. Series of elements of Gaussian Markov sequences

Transcription

1 On the almost sure convergence of series of elements of Gaussian Markov sequences Runovska Maryna NATIONAL TECHNICAL UNIVERSITY OF UKRAINE "KIEV POLYTECHNIC INSTITUTE" January 2011

2 Generally this talk deals with the investigation of asymptotical behavior of zero-mean Gaussian Markov sequences. It is well-known that a zero-mean Gaussian Markov sequence is the first-order autoregressive sequence. Different autoregressive sequences, in particular, their asymptotic properties, were studied very widely. For example, such a sequences were considered in the works of V.V. Buldygin (1978), M. Arato (1982), A.P. Korostelev (1984), V.V. Buldygin and S.A. Solntsev (1989, 1997), V.A. Koval (1991), A.Ya. Dorogovtsev (1992), A.M. Iksanov (2007) and others.

3 As to Gaussian Markov sequence, the necessary and sufficient conditions for the almost sure (a.s.) convergence to zero were found in the work of V.V. Buldygin (1978). Moreover, the necessary and sufficient conditions for the a.s. convergence and a.s. boundedness for such a sequences were investigated. Further these conditions were generalized on a multi-dimensional case. Moreover, the necessary and sufficient conditions for the convergence a.s. to zero of Gaussian m-markov sequences of random variables were studied (V.V. Buldygin and S.A. Solntsev, 1997). Continuing the investigations of asymptotical properties of Gaussian Markov sequences, the problem of finding the necessary and sufficient conditions for the convergence a.s. of a series consisting of elements of such a sequence arises.

4 The main problem Introduction Thus, consider a finite-dimensional Euclidian space R d, d 1. Let (X k ) = (X k, k 1) be a zero-mean Gaussian Markov sequence in the space R d, i.e. the sequence which obeys the system of stochastic recurrence equations X 1 = D 1 Γ 1, X k = C k X k 1 + D k Γ k, k 2, where (C k ) and (D k ) are nonrandom sequences of real squared matrices of order d, and (Γ k ) is a sequence of jointly independent standard Gaussian random vectors in the space R d. For the Gaussian Markov sequence (X k ) consider the random series X k. k=1 The main purpose of this paper is to find the necessary and sufficient conditions for the convergence a.s. of this series.

5 Another important problem Let us emphasize that there is also another important reason to find a criterion for the convergence a.s. of series k=1 X k. Indeed, let (ξ k ) be the Gaussian m-markov sequence of random variables, i.e. the sequence which obeys the m-th order system of recurrence equations: ξ 1 m =... = ξ 1 = ξ 0 = 0, ξ k = b k1 ξ k 1 + b k2 ξ k b km ξ k m + β k γ k, k 1, where (β k ) is non-negative nonrandom real sequence, (b kj ; 1 j m, k 1) is a nonrandom real array, and (γ k ) is a standard Gaussian sequence. For the sequence (ξ k ) consider the random series ξ k. k=1 The necessary and sufficient conditions for the convergence a.s. of series k=1 ξ k were not investigated before.

6 The connection between two problems Consider the Gaussian m-markov sequence ξ 1 m =... = ξ 1 = ξ 0 = 0, ξ k = b k1 ξ k 1 + b k2 ξ k b km ξ k m + β k γ k, k 1, and for k 1 set ξ k ξ X k = k 1, Γ... k = ξ k m+1 γ k , D k = β k , C k = b k1 b k2... b km Then instead of the recurrence relation of the m-th order in the space R one can consider the recurrence relation of the first order in the space R m..

7 Namely, X 1 = D 1 Γ 1, X k = C k X k 1 + D k Γ k, k 2. Obviously, the sequence (X k ) is a zero-mean Gaussian Markov sequence in the space R m. Therefore the random series k=1 ξ k for the Gaussian m-markov sequence ξ k converges a.s. if and only if the series k=1 X k converges a.s. in the norm of the space R m.

8 Main results Introduction Thus, in the sequel we consider a Euclidian space R d, d 1, with inner product (X, Y) and endowed with the norm X = (X, X), X, Y R d. Let be a matrix norm in the space R d, for any matrix A = (a ij ) d interpreted as follows i,j=1 ( d 1/2. A = ai,j) 2 i,j=1

9 Let (X k ) be a zero-mean Gaussian Markov sequence in the space R d : X 1 = D 1 Γ 1, X k = C k X k 1 + D k Γ k, k 2, where (C k ) and (D k ) are nonrandom sequences of real squared matrices of order d, and (Γ k ) is a sequence of jointly independent standard Gaussian random vectors in the space R d. Let us find the necessary and sufficient conditions for the convergence a.s. of series X k. k=1 Denote by R the class of all monotone sequences of positive integers which increase to infinity.

10 Theorem 1 The series k=1 X k converges a.s. if and only if the following three conditions hold: 1) for any k 1 the matrix series ( k+1 ) l=1 j=k+l C j D k, is convergent in the matrix norm of the space R d ; 2) k=1 Q(, k) 2 <, where Q(, k) = D k + ( k+1 ) l=1 j=k+l C j D k ; 3) for any ε > 0 and all the sequences (m j ) from the class R, one has { ε } exp mj+1 j=1 k=m j +1 Q(m <, j+1, k) 2 D k + ( n k k+1 ) l=1 j=k+l C j D k, 1 k n 1; where Q(n, k) = D k, k = n; O, k > n,

11 Now consider the case d = 1. This case is studied in the work of M.K. Runovska, Let (ζ k ) be a zero-mean Gaussian Markov sequence of random variables ζ 1 = β 1 γ 1, ζ k = α k ζ k 1 + β k γ k, k 2, where (α k ) is nonrandom real sequence, (β k ) is non-negative nonrandom real sequence, and (γ k ) is a standard Gaussian sequence. The following criterion for the convergence a.s. of series holds true. k=1 ζ k

12 ζ 1 = β 1 γ 1, ζ k = α k ζ k 1 + β k γ k, k 2. Corollary 1 (M.K. Runovska, 2010) The series k=1 ζ k converges a.s. if and only if the following three conditions hold: ( 1) for any k 1 the series l=1 β ) k+l k j=k+1 α j converges; 2) k=1 (A(, ( k))2 <, where A(, k) = β k 1 + ) k+l l=1 j=k+1 α j, k 1; 3) for any ε > 0 and any sequence (m j ) R { ε } exp mj+1 j=1 k=m j +1 (A(m <, j+1, k)) 2 where ( β k 1 + ( n k k+l l=1 α j=k+1 j A(n, k) = β k, k = n; 0, k > n. )), 1 k n 1;

13 The criterion for the convergence a.s. of series k=1 X k for multi-dimensional Gaussian Markov sequence (X k ) enables to find the criterion for the convergence a.s. of series of elements of Gaussian m-markov sequence of random variables.

14 A Criterion for the convergence a.s. of a series of elements of Gaussian m-markov sequence Consider the sequence of random variables (ξ k ) which obeys the m-th order system of recurrence equations: ξ 1 m =... = ξ 1 = ξ 0 = 0, ξ k = b k1 ξ k 1 + b k2 ξ k b km ξ k m + β k γ k, k 1, where (β k ) is non-negative nonrandom real sequence, (b kj ; 1 j m, k 1) is a nonrandom real array, and (γ k ) is a standard Gaussian sequence. For the sequence (ξ k ) consider the random series k=1 and find a criterion for the convergence a.s. of this series. ξ k

15 Theorem 2 The random series k=1 ξ k converges a.s. if and only if the following three conditions hold: ( ) 1) for any k 1 the nonrandom series l=0 β k u (k+1) is k+l convergent, where (u (k) n, n k) is a nonrandom sequence which obeys the system of recurrence equations u (k+1) n = b n1 u (k+1) n 1 + b n2 u (k+1) n b nm u (k+1) n m, n k + 1, u (k+1) k (m 1) = u(k+1) =... = u(k+1) = 0, u (k+1) = 1; k (m 2) k 1 k 2) k=1 U2 k <, where U k = l=0 β k u (k+1), k 1; k+l 3) for any ε > 0 and all the sequences (m j ) from the class R, one has { ε } exp ( mj+1 mj+1 ) k 2β <. j=1 u (k+1) 2 k=m j +1 l=0 k+l k

16 Gaussian m-markov sequence with constant coefficients ξ 1 m =... = ξ 1 = ξ 0 = 0, ξ k = b 1 ξ k 1 + b 2 ξ k b m ξ k m + β k γ k, k 1, where b j, j = 1, 2,..., m, are some real constants, (β k ) is non-negative nonrandom real sequence such that there are non-equal to zero elements among β k s. Corollary 2 The random series k=1 ξ k converges a.s. if and only if the following two conditions hold: 1) max λ k < 1, where λ i, i = 1, 2,..., s, s m, are different 1 k s roots of the equation λ m (b 1 λ m 1 + b 2 λ m b m 1 λ + b m ) = 0; 2) k=1 β2 k <.

17 Example (m=2) Introduction ξ 1 = ξ 0 = 0, ξ k = aξ k 1 + bξ k 2 + β k γ k, k 1, where a and b are some real constants, and (β k ) is nonrandom non-negative real sequence. The random series k=1 ξ k converges a.s. if and only if the following two conditions hold: 1) 1 < b < 1 a ; 2) k=1 β2 k <. b a

18 M. Arato Linear stochastic systems with constant coefficients: a statistical approach, Springer-Verlag: Berlin-Heidelberg, p. V.V. Buldygin The strong laws of large numbers and convergence to zero of Gaussian sequences, Probab. Theory and Math. Stat P (in Russian) V.V. Buldygin, S.A. Solntsev, Functional methods in summability problems of random variables, Naukova dumka, Kiev, p. (in Russian) V.V. Buldygin and S. A. Solntsev, Asymptotic behavior of linearly transformed sums of random variables, Kluwer Academic Publishers, Dordrecht, V.V. Buldygin, M.K. Runovska On the convergence of series of autoregressive sequences, Theory of stochastic processes (31) P.7 14.

19 V.V. Buldygin, M.K. Runovska On the convergence of series of autoregressive sequences in Banach spaces, Theory of stochastic processes (32) P M.K. Runovska Convergence of series of elements of Gaussian Markov sequences, Probab. Theory and Math. Stat., 2010, No. 83, P (in Ukrainian) A.Ya. Dorogovtsev Periodic and stationary regulations of infinite-dimensional determined and stochastic dynamical systems, Vyscha shkola, Kiev, p. (in Russian) A.M. Iksanov Random series of special kind, branchy random walk and self-decomposability, Kiev: Kti Print, p. (in Ukrainian) V.A. Koval Asymptotical behavior of solutions of stochastic recurrence equations in space R d, Ukr. Math. Journal, 1991, No. 6(43), P (in Russian) A.P. Korostelev Stochastic recurrence procedures, Local properties, Nauka, Moscow, p. (in Russian)