[I2], [IK1], [IK2], [AI], [AIK], [INA], [IN], [IK2], [IA1], [I3], [IKP], [BIK], [IA2], [KB]

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1 (Akihiko Inoue) Graduate School of Science, Hiroshima University (Yukio Kasahara) Graduate School of Science, Hokkaido University Mohsen Pourahmadi, Department of Statistics, Texas A&M University 1, = ( ) ( ) von Neumann, =. [I1] [I2], [IK1], [IK2], [AI], [AIK], [INA], [IN], [IK2], [IA1], [I3], [IKP], [BIK], [IA2], [KB],, (Orthogonal Polynomials on the Unit Circle, OPUC). Barry Simon 2005 AMS OPUC 2 [Si1, Si2] OPUC Verblunsky, Verblunsky =.,, Verblunsky. Verblunsky CGMV ([CGMV]). Simon rigid (phase coefficients Nehari ).,, Bingham [KB]. 1

2 rigid (rigid Sarason [Sa2, Ch. X] ) rigid Sarason [Sa1] [N1] strongly outer 1 ([AIK], [INA], [IN] ) , [I1], [I2], [I3], [IK1], [IK2], [KB] 2013 rigid (= ) rigid 2 rigid T D C 1 p L p T ( Lebegue ) Lebesgue M := M d (C) d d V := C d Euclid. T M - f det f 0 a.e. L p M (Lp V ) T M - (V - ) Lebesgue. Hardy H p M : H p M {f := L p M 2π 0 } e inθ f(e iθ )dθ = 0 for n = 1, 2,. Hardy H p V M V. g H p M outer det g(z) 1 outer c = det g(0)/ det g(0) : ( 1 2π ) e iθ + z det g(z) = c exp 2π 0 e iθ z log det g(eiθ ) dθ (z D). 2 (d 2) outer g : D := {(f, f ) H 2 M H 2 M : f, f outer, f f = f f }. F H 1 M : F = ff, (f, f ) D (2.1) 2

3 (Helson Lowdenslager[HL] ). HL- F f HM 2 (f, f ) D f, F HM 1 U F F = U F F F = F F U F. HL- F = ff, : U F = f(f ) 1 = (f ) 1 f. (2.2) ( ) rigid. 2.1 ( rigid ). F HM 1 rigid : G HM 1 G = gg HL- U F = U G = G = fkf k M. rigid outer, rigid outer (1 ) : H 2 V := { f L 2 V 2π 0 } e inθ f(e iθ )dθ = 0 for n = 0, 1, 2, (f, f ) D F := ff. : (1) F rigid. (2) (g, g ) D, G = gg, U F = U G g = fc c M. (3) U F H 2 V H 2 V = {0}., rigid 1. X = (X j ) j Z, X j = (X j,1,, X j,d ), 0 d- X (purely non-deterministic, PND) full rank det w > 0 a.e., log det w L 1 w L 1 M w : E[Xj X 0 ] = 1 2π e ijθ w(θ)dθ (j Z) 2π 0 w = h h = h h. (2.3) 3

4 h h HM 2 outer (cf. [HL]), 1 (d=1) w = h 2, h = h. 2 h h 1 2 I R H I {X j,k : j I Z, k = 1,, d} L 2 (Ω, F, P ) 2.3. X (completely non-deterministic, CND) H (,0] H [1, ) = {0}. : X CND h(h ) 1 H 2 V H 2 V = {0} : (1) hh rigid. (2) (g, g ) D, h(h ) 1 = g(g ) 1 g = fc c M. (3) X CND. 2.2, 2.4 Levinson McKean [LM], Bloomfield et al. [BJH], [N2], CND 1. PND X, X 0, X 1,, X n 1 X n ˆX n, : ˆX n = X n 1 ϕ 1,n + X n 1 ϕ 2,n + + X 0 ϕ n,n. (2.4) n ϕ n,n. 1 (d = 1), {ϕ n,n } n=1 {X n }, OPUC w Verblunsky. v := h(0) h(0), v := h (0)h (0). 1- (phase coefficients) γ n : γ n := 1 2π e inθ h(h 2π ) 1 dθ. (2.5) 0 1 [BIK] [KB], ϕ n,n n ( ϕ n,k γ n ) [IK2] [I3]. 4

5 2.5. X CND, ϕ n,n : ϕ n,n = v 1 ( k=0 ) t γ n ( Γ n Γ n ) k e v (n N). (2.6) : γ n γ n := γ n+1 γ n+2, Γ n := γ n+1 γ n+2 γ n+3 γ n+2 γ n+3 γ n+4 γ n+3 γ n+4 γ n+5, e := id o o. References [AI] [AIK] [BIK] [BJH] V. V. Anh and A. Inoue, Financial markets with memory I: Dynamic models, Stochastic Anal. Appl. 23 (2005), V. V. Anh, A. Inoue and Y. Kasahara, Financial markets with memory II: Innovation processes and expected utility maximization, Stochastic Anal. Appl. 23 (2005), N. H. Bingham, A. Inoue and Y. Kasahara, An explicit representation of Verblunsky coefficients, Statist. Probab. Lett., 82 (2012), P. Bloomfield, N. P. Jewell and E. Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), [CGMV] M. J. Cantero, F. A. Grünbaum, L. Moral and L. Velázquez, Matrix-valued Szeg o polynomials and quantum random walks, Commun. Pure Appl. Math. 58 (2010), [HL] [I1] [I2] [I3] [IA1] [IA2] [IK1] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables. Acta Math. 99 (1958), ; II. Acta Math. 106 (1961) A. Inoue, Asymptotics for the partial autocorrelation function of a stationary process, J. Anal. Math. 81 (2000), A. Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes, Ann. Appl. Probab. 12 (2002), A. Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), A. Inoue and V. V. Anh, Prediction of fractional Brownian motion-type processes, Stoch. Anal. Appl. 25 (2007), A. Inoue and V. V. Anh, Prediction of fractional processes with long-range dependence, Hokkaido Math. J. 41 (2012), A. Inoue and Y. Kasahara, Partial autocorrelation functions of fractional ARIMA processes with negative degree of differencing, J. Multivariate Anal. 89 (2004),

6 [IK2] [IKP] [IN] [INA] [KB] A. Inoue and Y. Kasahara, Explicit representation of finite predictor coefficients and its applications, Ann. Statist. 34 (2006), A. Inoue, Y. Kasahara and P. Phartyal, Baxter s inequality for fractional Brownian motion-type processes with Hurst index less than 1/2, Statist. Probab. Lett. 78 (2008), A. Inoue and Y. Nakano, Optimal long-term investment model with memory, Appl. Math. Optim. 55 (2007), 55 (2007), A. Inoue, Y. Nakano and V. V. Anh, Linear filtering of systems with memory and application to finance, J. Appl. Math. Stoch. Anal. (2006), Art. ID 53104, 26 pp. Y. Kasahara and N. H. Bingham, Verblunsky coefficients and Nehari sequences, Trans. Amer. Math. Soc., electronically published in [LM] N. Levinson and H. P. McKean, Weighted trigonometrical approximation on R 1 with application to the germ field of a stationary Gaussian noise, Acta Math. 112 (1964), [N1] T. Nakazi, Exposed points and extremal problems in H 1, J. Funct. Anal. 53 (1983), [N2] T. Nakazi, Kernels of Toeplitz operators, J. Math. So. Japan 38 (1986), [Sa1] D. Sarason, Exposed points in H 1, Oper. Theory Adv. Appl. 41 (1989), [Sa2] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Wiley, [Si1] [Si2] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory., AMS, B. Simon, S., Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory., AMS,