The Annals of Mathematical Statistics

A Limit Theorem for Random Interval Sampling of a Stochastic Process

Richard F. Gebhard

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A time series is the realization of a stochastic process. An estimate of the process average can be taken as the equally weighted average of observed values taken from the time series. Normally, the sequence of observations consists of observations taken at equal intervals of time. In certain applications, it is more appropriate to consider a sequence of observations spaced by intervals which are randomly and independently chosen from an exponential distribution. Let $\lbrack x_t, t \varepsilon T\rbrack$ be a real stochastic process that takes on values in a space $X, T$ being restricted to the real line. Let $\Omega$ be a space of points $\omega, F_T = \beta(x_t, t \varepsilon T)$ be the Borel field of $\omega$-sets generated by the class of sets of the form $\lbrack x_t(\omega) \varepsilon A\rbrack$ where $t \varepsilon T$ and $A$ is any Borel set, $P$ be the probability measure of $F_T$-sets, $\alpha$ be the Borel field of $X$-sets and for every real $t, x_t(\cdot)$ be a function from $\Omega$ to $X$ such that $\lbrack x_t(\omega) \varepsilon A\rbrack$ is an $F_T$-set for $A \varepsilon \alpha$. Assume that the process satisfies, for all $t \varepsilon T$, the conditions \begin{align*}\tag{1.1}\quad E\{x_t\} &= \bar x \quad \text{exists and is independent of} \quad t \\ \tag{1.2}E\{(x_t - \bar x^2\} &= \operatorname{var} x \quad \text{exists and is independent of} \quad t \\ \tag{1.3} E\{x_t - \bar x) (x_{t + \tau} - \bar x)\} &= r \quad (\tau) \operatorname{var} x \quad \text{depends only on} \quad \tau \quad \text{and} \\ r \quad (\tau) \text{is harmonizable}. \end{align*} A stochastic process satisfying Conditions (1.1) to (1.3) is often referred to as being stationary in the wide sense [2]. \begin{equation*}\tag{1.4}\lim_{\tau\rightarrow\infty} \tau r(\tau) \rightarrow 0.\end{equation*} Let $\{\xi\}$ be a measurable space of mutually independent random variables $\xi_i$ identically distributed with probability $p$ of having value 1 and $(1 - p)$ of having value 0, all independent of the $x_t$-process. The sample space $\Omega$ contains all possible realizations of the stochastic process. Given a scheme for observing the values of $x_t$ in time and some form of estimate of process mean from the observed values, expected values of such an estimate and of its square can be determined. A measure of the efficiency of the sampling scheme may be defined as the ratio of the variance of the estimator to that which would be obtained if the observations taken were independent of each other. The limit in probability of this measure of sampling efficiency for observations spaced by exponentially distributed mutually independent intervals is obtained simply in terms of the mean observation rate and the spectral density component at zero frequency. The limit theorem considered is a weak version. By augmenting Condition (1.4) with relations between sample size $N$, mean observation rate $\lambda$, length of time series $T$ and sufficiently rapid decay of $r(\tau)$ with $\tau$, sharper probability statements can be made.

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Ann. Math. Statist., Volume 35, Number 2 (1964), 866-868.

First available in Project Euclid: 27 April 2007

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Gebhard, Richard F. A Limit Theorem for Random Interval Sampling of a Stochastic Process. Ann. Math. Statist. 35 (1964), no. 2, 866--868. doi:10.1214/aoms/1177703587.

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