×

zbMATH — the first resource for mathematics

A pairwise independent stationary stochastic process. (English) Zbl 0569.60041
Properties of pairwise independent strictly stationary stochastic processes \(X_ n\), \(n\in {\mathbb{Z}}\), with \(P\{X_ n=1\}=P\{X_ n=-1\}=\) are considered. It is shown e.g. that always \(| E(X_ 1X_ 2X_ 3)| \leq\); and there exists one and only one \(X_ n\), \(n\in {\mathbb{Z}}\), having the properties (i) \(X_ n\in \{-1,1\}\), (ii) \(E(X_ n)=0\), (iii) \(X_ n\), \(n\in {\mathbb{Z}}\), is pairwise independent and stationary, (iv) \(\{X_ 1,...,X_{n-1}\}\) and \(X_ n\) are independent for all \(n>1\), (v) \(E(X_ 1X_ 2X_ 3)=\).
Reviewer: H.Niemi

MSC:
60G10 Stationary stochastic processes
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Breiman, L, Probability, (1968), Addison-Wesley Publishing Co., Reading, MA · Zbl 0174.48801
[2] Chung, K.L, A course in probability theory, (1974), Academic Press New York · Zbl 0159.45701
[3] Joffe, A, On a set of almost determinable k-independent random variables, Ann. prob., 2, 161-162, (1974) · Zbl 0276.60005
[4] Lancaster, H.O, Pairwise statistical independence, Ann. math. statist., 32, 1313-1317, (1965) · Zbl 0131.18105
[5] Lévy, P, Exemple de processus pseudo-markoviens, C.R. acad. sci. Paris, 288, 2004-2006, (1949) · Zbl 0041.25201
[6] O’Brien, G.L, Pairwise independent random variables, Ann. prob., 8, 170-175, (1980) · Zbl 0426.60011
[7] Robertson, J, The mixing properties of certain processes related to Markov chains, Math. sys. th., 7, 39-43, (1973) · Zbl 0256.60054
[8] Robertson, J, A spectral representation of the states of a measure preserving transformation, Z. wahrsch., 27, 185-194, (1973) · Zbl 0281.60037
[9] Rosenblatt, M; Slepian, D, Nth order Markov chains with every N variables independent, J. soc. indust. appl. math., 10, 537-549, (1962) · Zbl 0154.43103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.