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1959•Proceedings of the American Mathematical SocietyOpen access

On a question of Paul Lévy

R. V. Chacon

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Abstract

The terms of such a transition matrix are called transition functions (of the underlying Markov chain). Paul Levy [2, ?111.4] asks whether each such transition function, which is not measurable, must be a transition function of a Markov chain obtained by combining a Markov chain having measurable transition functions with one having a transition matrix of nonmeasurable functions which take only the values zero or one. Levy [2, Theoreme II. 8.2] has shown that if {pjj(t)}, i, j = 1, 2, ,is a transition matrix of measurable transition functions, then limt,, pij(t) exists, and

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The terms of such a transition matrix are called transition functions (of the underlying Markov chain). Paul Levy [2, ?111.4] asks whether each such transition function, which is not measurable, must be a transition function of a Markov chain obtained by combining a Markov chain having measurable transition functions with one having a transition matrix of nonmeasurable functions which take only the values zero or one. Levy [2, Theoreme II. 8.2] has shown that if {pjj(t)}, i, j = 1, 2, ,is a transition matrix of measurable transition functions, then limt,, pij(t) exists, and

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Available abstract

The terms of such a transition matrix are called transition functions (of the underlying Markov chain). Paul Levy [2, ?111.4] asks whether each such transition function, which is not measurable, must be a transition function of a Markov chain obtained by combining a Markov chain having measurable transition functions with one having a transition matrix of nonmeasurable functions which take only the values zero or one. Levy [2, Theoreme II. 8.2] has shown that if {pjj(t)}, i, j = 1, 2, ,is a transition matrix of measurable transition functions, then limt,, pij(t) exists, and

Key concepts: Markov chain, Stochastic matrix, Transition (genetics), Chain (unit), Mathematics, Function (biology), Transition rate matrix, Matrix (chemical analysis)

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