We consider the finite-past predictor coefficients of stationary time series, and establish an explicit representation for them, in terms of the MA and AR coefficients. The proof is based on the alternate applications of projection operators associated with the infinite past and the infinite future. Applying the result to long memory processes, we give the rate of convergence of the finite predictor coefficients and prove an inequality of Baxter-type

We develop a prediction theory for a class of processes with stationary increments. In particular, we prove a prediction formula for these processes from a finite segment of the past. Using the formula, we prove an explicit representation of the innovation processes associated with the stationary increments processes. We apply the representation to obtain a closed-form solution to the problem of expected logarithmic utility maximization for the financial markets with memory introduced by the first and second authors.

For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter's inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.2010 Mathematics Subject Classification. Primary 60G25; secondary 62M20, 62M10.

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Let {Xn : ∈Z} be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|~|d|/n as n→∞. This extends the previous result for the case 0

Abstract. For a nonnegative integrable weight function w on the unit circle T , we provide an expression for p = 2, in terms of the series coefficients of the outer function of w, for the weighted L p distance inf f T |1 − f | p wdµ, where µ is the normalized Lebesgue measure and f ranges over trigonometric polynomials with frequencies in [{. . . , −3, −2, −1}\{−n}]The problem is open for p = 2.

A SiC thin-film thermistor for high-temperature use has been developed by using rf-sputtered SiC thin films. This thermistor can be used for industrial and consumer use within an operating temperature range of -100 to 450 degrees C. By using SiC thin films, the thermistor maintains high electrical stability. The resistance change is less than 3% after exposure to heat at 400 degrees C for 2000 h. In addition, the film growth technique made possible the production of a high-accuracy thermistor, i.e., thermistor coefficient < +/-0.5%, thermistor resistance < +/-1.5%.

We are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle. In general, the Szegö function is determined by its modulus, while the condition in question is that it is also determined by its argument, or in terms of the function theory, that the square of the Szegö function is rigid. In prediction theory, this is known as a spectral characterization of complete nondeterminacy for stationary processes, studied by Bloomfield, Jewel and Hayashi (1983) going back to a small but important result in the work of Levinson and McKean (1964). It is also related to the cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem, and there is a one-to-one correspondence between the Verblunsky coefficients and the negatively indexed Fourier coefficients of the phase factor of the Szegö function, which we call a Nehari sequence. We present some fundamental results on the correspondence, including extensions of the strong Szegö and Baxter's theorems.