The Annals of Mathematical Statistics

Random Functions Satisfying Certain Linear Relations, II

S. G. Ghurye

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Abstract

We consider here another aspect of a problem mentioned in a previous paper [2]. We shall be concerned with one-dimensional, real-valued random functions (r.f.) $X(t)$, defined for all $t$ and such that any sample taken at equidistant $t$-points satisfies a linear relation which is an analogue of one or other of the stochastic difference relations which are used for the analysis of discrete-parameter time-series. More specifically, we assume that there exist $k$ continuous and real-valued functions $\alpha_1(h), \cdots, \alpha_k(h) \text{of} h \geqq 0$ such that for any $h > 0$ and any $t$, the sequence \begin{equation*}\tag{1}\{X(t + \lbrack n + k \rbrack h) + \alpha_1(h)X(t + \lbrack n + k - 1 \rbrack h) + \cdots + \alpha_k(h)X(t + nh)\}, n = 0, \pm 1, \cdots,\end{equation*} satisfies certain conditions about independence or noncorrelation. In Section 1, we consider hypotheses concerning correlation, and find that the functions $\alpha_j(h)$ are restricted to certain forms. We also find that the assumption of zero serial correlations in the sequence (1) for all $h > 0$ implies that $X(t)$ is deterministic. In Section 2, we consider hypotheses concerning independence, and find the functions $\alpha_j(h)$ to be restricted as before.

Article information

Source
Ann. Math. Statist., Volume 26, Number 1 (1955), 105-111.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728597

Digital Object Identifier
doi:10.1214/aoms/1177728597

Mathematical Reviews number (MathSciNet)
MR67393

Zentralblatt MATH identifier
0064.38104

JSTOR
links.jstor.org

Citation

Ghurye, S. G. Random Functions Satisfying Certain Linear Relations, II. Ann. Math. Statist. 26 (1955), no. 1, 105--111. doi:10.1214/aoms/1177728597. https://projecteuclid.org/euclid.aoms/1177728597


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See also

  • Part I: S. G. Ghurye. Random Functions Satisfying Certain Linear Relations. Ann. Math. Statist., Volume 25, Number 3 (1954), 543--554.