(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],, 2011. (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
rigid (rigid Sarason [Sa2, Ch. X] ) rigid Sarason [Sa1] [N1] strongly outer 1 ([AIK], [INA], [IN] ) 2 2005 2011, [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
(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,. 2.2. (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
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}. 2.2 2.4. 3 : (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
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), 275 300. V. V. Anh, A. Inoue and Y. Kasahara, Financial markets with memory II: Innovation processes and expected utility maximization, Stochastic Anal. Appl. 23 (2005), 301 328. N. H. Bingham, A. Inoue and Y. Kasahara, An explicit representation of Verblunsky coefficients, Statist. Probab. Lett., 82 (2012), 403 410. P. Bloomfield, N. P. Jewell and E. Hayashi, Characterizations of completely nondeterministic stochastic processes, Pacific J. Math. 107 (1983), 307 317. [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), 464 507. [HL] [I1] [I2] [I3] [IA1] [IA2] [IK1] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables. Acta Math. 99 (1958), 165-202; II. Acta Math. 106 (1961) 175 213. A. Inoue, Asymptotics for the partial autocorrelation function of a stationary process, J. Anal. Math. 81 (2000), 65 109. A. Inoue, Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes, Ann. Appl. Probab. 12 (2002), 1471 1491. A. Inoue, AR and MA representation of partial autocorrelation functions, with applications, Probab. Theory Related Fields 140 (2008), 523 551. A. Inoue and V. V. Anh, Prediction of fractional Brownian motion-type processes, Stoch. Anal. Appl. 25 (2007), 641 666. A. Inoue and V. V. Anh, Prediction of fractional processes with long-range dependence, Hokkaido Math. J. 41 (2012), 157 183. A. Inoue and Y. Kasahara, Partial autocorrelation functions of fractional ARIMA processes with negative degree of differencing, J. Multivariate Anal. 89 (2004), 135 147. 5
[IK2] [IKP] [IN] [INA] [KB] A. Inoue and Y. Kasahara, Explicit representation of finite predictor coefficients and its applications, Ann. Statist. 34 (2006), 973 993. 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), 2889 2894. A. Inoue and Y. Nakano, Optimal long-term investment model with memory, Appl. Math. Optim. 55 (2007), 55 (2007), 93 122. 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 2013. [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), 99 143. [N1] T. Nakazi, Exposed points and extremal problems in H 1, J. Funct. Anal. 53 (1983), 224 230. [N2] T. Nakazi, Kernels of Toeplitz operators, J. Math. So. Japan 38 (1986), 607 616. [Sa1] D. Sarason, Exposed points in H 1, Oper. Theory Adv. Appl. 41 (1989), 485 496. [Sa2] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Wiley, 1994. [Si1] [Si2] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory., AMS, 2005. B. Simon, S., Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory., AMS, 2005. 6