Sampling Theorems for Non-Stationary Random ProcessesAuthor(s): Alan J. LeeSource: Advances in Applied Probability, Vol. 11, No. 2 (Jun., 1979), pp. 299-300Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426830 .

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Canberra, Australia, 6-10 July 1978 299

the transient evaluated at the fundamental frequencies 24kc/T are approxi- mately constant over narrow bands of frequencies. This has implications for the estimation of both time domain and frequency domain models, but the particu- lar example to be discussed in detail is the estimation of the velocity of a transient signal received at an array of sensors (Cameron and Hannan (1979)). Such data have, in the past, been treated by estimates designed for stationary processes.

A suite general model for transients may be given which quantifies the provision mentioned initially. Using this model a strong law of large numbers and a central limit theorem may be proved for an estimate of velocity derived using the smoothness of the Fourier coefficients. The methods have been applied to seismic data.

Reference

CAMERON, M. A. AND HANNAN, E. J. (1979) Transient signals. Biometrika 66 (2).

Sampling theorems for non-stationary random processes

ALAN J. LEE, University of Auckland

A stationary random process x, is band-limited if its spectral measure is concentrated on a finite interval [- w, w]. Such processes satisfy the so-called 'sampling theorem' (see e.g. [1])

(1) = Xh sin rh-l(t - nh)

n rn -'(t - nh)

for h-l <2w, and the Hilbert space H(x) generated by the r.v.'s x, e R, is also generated by the 'samples' x,, n E Z; i.e. the process is 'linearly determined by its samples'. It is not necessary that a process be band-limited in order that it be linearly determined by its samples; a necessary and sufficient condition for this is that the spectral measure of x has a support whose translates by nh-1, n E Z/are all disjoint, and if this condition is satisfied, sampling theorems similar to (1) are possible under mild restrictions. (See [2].)

All of these ideas can be generalized to random processes that are not stationary, provided the quantities Var x, are bounded for all t by some polynomial. A definition of a band-limited non-stationary process is given, and such processes are shown to satisfy sampling theorems similar to (1). A condition sufficient for a process to be linearly determined by its samples is also given, and some sampling theorems for non-band-limited processes discussed.

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300 CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

References

[11 BEUTLER, F. J. (1961) Sampling theorems and bases in Hilbert space. Information and Control 4, 97-117.

[2] LLOYD, S. P. (1959) A sampling theorem for stationary stochastic processes. Trans. Amer. Math. Soc. 92, 1-12.

Multivariate spectral estimation in the presence of frequency-dependent time delays

P. J. THOMSON, Massey University, Palmerston North

A vector stationary time series X(t) is considered where the frequency components that make up the spectral representation of Xj(t) have undergone time delays that, in general, depend on j and the frequency concerned. A situation such as this will arise, for example, if a signal is transmitted through a dispersive medium so that the speed of propagation of the signal is frequency dependent. A procedure for estimating these time delays together with the spectra and cross spectra of X(t) is described and the various asymptotic properties of the estimators discussed.

References

HAMON, B. V. AND HANNAN, E. J. (1974) Spectral estimation of time delay for dispersive and non-dispersive systems. Appl. Statist. 23, 134-142.

HANBAN, E. J. AND THOMSON, P. J. (1973) Estimating group delay. Biometrika 60, 241-253. KNAPP, C. H. AND CARTER, G. C. (1977) Estimation of time delay in the presence of source or

receiver motion. J. Acoust. Soc. Amer. 61, 1545-1549.

III. Inference

Asymptotic theory of inference for a certain class of stochastic processes

I. V. BASAWA, La Trobe University, Bundoora, Victoria

Let X" = (X1, X2, - - - , X,) be a realization from a stochastic process with the joint density p,(xn; 0); 06 e , an open interval of the real line. The score function, S,(0)=dlogp,(Xn"; )/dO, and the information function, 4,(0)= - d2 log p,(X"; O)/dO2 play an important role in the asymptotic theory of

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