Random Functions Satisfying Certain Linear Relations

Abstract

A hypothesis often made about a sequence of real-valued r.v. (random variables), $\{X_n, n = 0, \pm 1, \pm 2, \cdots\}$ is that there exist certain real constants $\alpha_1, \alpha_2, \cdots, \alpha_k$ such that if we write \begin{equation*}\tag{1} Y_n = X_n + \alpha_1X_{n-1} + \cdots + \alpha_kX_{n-k},\end{equation*} then $\{Y_n\}$ is a sequence of independent r.v. Now, very often in practice, the observed sequence $\{X_n\}$ consists in observations made at equidistant $t$-points on a stochastic process with a continuous parameter $t$. Restricting our attention to the case $k = 1$, we then have a r.f. (random function) $X(t)$, defined for all $t$ in an interval, with the following property: There exist a value of $t (t_0$, say), and $h > 0$, and a real number $\alpha$ such that a hypothesis of the type mentioned is satisfied by the sequence $\{Y(t_0, h; n), n = 0, \pm1, \pm2, \cdots\}$, where \begin{equation*}\tag{2} Y(t_0, h; n) = X(t_0 + nh) - \alpha X(t_0 + \lbrack n - 1\rbrack h).\end{equation*} But if such a hypothesis is true for one value of $h$, it is not necessarily so for some other value; and we have to make the additional assumption that the particular length of the $t$-intervals with which we are concerned is precisely the one for which the hypothesis holds. This assumption may not be reasonable in every case, Instead we may wish to work with a hypothesis similar to the above, but which holds for all positive $h$ in some interval. In this paper, we investigate the existence and form of random functions satisfying a hypothesis of this type. Section 1 contains a statement of the problem and some simple results. It turns out that any random function possessing the required property can be expressed as the product of an exponential function of $t$ and a r.f. with independent increments. Section 2 deals with the limit in distribution of a sequence of Stieltjes approximating sums involving a r.f. with independent increments. Finally, in Section 3, these results are applied to the problem under investigation, and further possibilities are investigated. It must be noted that, in this investigation, we are concerned only with results in distribution. That is to say, we are dealing throughout only with parametric families of probability laws and talking in terms of r.f. merely for convenience.