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ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1301518
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Abstract: The distributions of two--block--factors $(f (\eta_{i},\eta_{i+1}); \, i \geq 1)$ arising from i.i.d. sequences $(\eta_{i}; \, i \geq 1)$ are observed to coincide with the distributions of the superdiagonals $(\zeta_{i,i+1}; \, i \geq 1)$ of jointly exchangeable and dissociated arrays $(\zeta_{i,j}; \, i, j \geq 1)$. 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.

References [Enhancements On Off] (What's this?)

  • 1. Aaronson J., Gilat D. and Keane M.S., On the structure of $1$--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 $k$--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

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