On two--block--factor sequences

and one--dependence

Author:
F. Matús

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1237-1242

MSC (1991):
Primary 60G10; Secondary 60J10, 60E15

DOI:
https://doi.org/10.1090/S0002-9939-96-03094-8

MathSciNet review:
1301518

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Abstract | References | Similar Articles | Additional Information

Abstract: The distributions of two--block--factors arising from i.i.d. sequences are observed to coincide with the distributions of the superdiagonals of jointly exchangeable and dissociated arrays . An inequality for superdiagonal probabilities of the arrays is presented. It provides, together with the observation, a simple proof of the fact that a special one--dependent Markov sequence of Aaronson, Gilat and Keane (1992) is not a two--block factor.

**1.**Aaronson J., Gilat D. and Keane M.S.,*On the structure of --dependent Markov shifts*, J. Theoretical Probab.**5**(1992), 545--561. MR**93i:60123****2.**Aaronson J., Gilat D., Keane M.S. and de Valk V.,*An algebraic construction of a class of one--dependent processes*, Ann. Probab.**17**(1989), 128--143. MR**89m:60084****3.**Aldous D.J.,*Representations for partially exchangeable arrays of random variables*, J. Multivariate Anal.**11**(1981), 581--598. MR**82m:60022****4.**Aldous D.J.,*Exchangeability and related topics*, Springer Lecture Notes in Math., No. 1117, 1985, 1--198. MR**88d:60107****5.**Burton R.M., Goulet M. and Meester R.W.J.,*On 1--dependent processes and --block--factors*, Ann. Probab.**21**(1993), 2157--2168. MR**94j:60072****6.**Eagleson G.K. and Weber N.C.,*Limit theorems for weakly exchangeable arrays*, Math. Proc. Cambridge Phil. Soc.**84**(1978), 123--130. MR**58:18670****7.**Gandolfi A., Keane M.S. and de Valk V.,*Extremal two--correlations of two--valued stationary one--dependent processes*, Probab. Theory Related Fields**80**(1989), 475--480. MR**90c:60012****8.**Hoover D.N.,*Row--column exchangeability and a generalized model for probability*. In: Koch G., Spizzichino F. (eds). Exchangeability in Probability and Statistics, North--Holland, Amsterdam, 1982, 281--291. MR**84b:60016****9.**Ibragimov I.A. and Linnik Y.V.,*Independent and stationary sequences of random variables,*Nauka, Moscow, 1965 (English translation: Wolters Noordhoff, Groningen, 1971). MR**48:1287****10.**Kallenberg O.,*Some new representations in bivariate exchangeability*, Probab. Theory Related Fields**77**(1988), 415--455. MR**89a:60101****11.**de Valk V.,*One--dependent processes: two--block--factors and non--two--block--factors*, CWI Tract 85, Amsterdam, 1994. CMP**94:10**

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Additional Information

**F. Matús**

Affiliation:
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Prague, Czech Republic

Email:
matus@utia.cas.cz

DOI:
https://doi.org/10.1090/S0002-9939-96-03094-8

Keywords:
$m$--dependence,
block--factors,
stationary sequences,
partially exchangeable arrays,
Markov chains,
weak topology,
superdiagonal

Received by editor(s):
February 24, 1994

Communicated by:
Richard T. Durrett

Article copyright:
© Copyright 1996
American Mathematical Society