Motivated by the pulse train stochastic process in the communication theory, a class of nonstationary linear stochastic processes is defined in terms of the stochastic integral with respect to nonorthogonal random measure. A general form of so called Bennett's formula which had been known only for pulse trains is obtained for the processes, and some concrete examples which sometimes appear in the applied field are given. Weak and strong lows of large numbers for the processes and asymptotic properties of the mean periodogram for them are also discussed in the paper.

Some properties of periodically correlated stochastic processes such as the mean ergodicity and asymptotic befavior of the periodgram of the processes are studied in this paper. A periodically correlated process (PC-process) is also called periodic nonstationary process, cyclo stationary process and even periodic stationary process although the process may not be stationary, and these are mainly studied as models of signals especially as pulse trains in the communication theory. Although many of interesting results obtained so far are important from the theoretical point of view, they are rather intuitively derived. Therefore we reformulate the process in a rigorous manner, introduce the spectral representation of it when the process is harmonizable in Loève's sense and study mainly about the mean ergodic properties and the limiting behavior of the mean periodgram of the process. Furthermore we study some pulse train processes as particular examples and we show simple examples of nonharmonizable PC-processes which had been thought of as unusual. We point out that by such a theoretical treatment, the position of PC-processes in the theory of nonstatonary processes will be better recognizable and the results obtained here will be useful as the foundation of practical time series and signal analysis in the communication and information theories.

This paper proves a general sampling theorem, which is an extension of Shannon's classical theorem. Let o be a closed subspace of square integrable functions and call o a signal space. The main aim of this paper is giving a necessary and sufficient condition for unique existence of the sampling basis {Sn}o without band-limited assumption. Using the general sampling theorem we rigorously discuss a frequency domain treatment and a general signal space spanned by translations of a single function. Many known sampling theorems in signal spaces, which have applications for multiresolution analysis in wavelets theory are corollaries of the general sampling theorem.

A necessary and sufficient condition of the ergodicity for a class of Gaussian periodically correlated stochastic processes is given in this paper. Periodically correlated processes are also called cyclostationary processes in wide sense, and these are mainly studied as models of signal processes in the communication theory and as models of time series of practical data of stochastic phenomena in some periodical environment. On the line of Boyles and Gardner's study(10) on cycloergodicity of certain class of nonstationary processes of discrete time parameter, we reformulate such ergodicity of cyclostationary processes in strict sense in continuous time parameter case, and prove a ergodic theorem. Futhermore, we apply it to Gaussian periodically correlated process, and discuss almost sure limiting behavior of a sample periodogram of the process.