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Transactions of the American Mathematical Society

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Zero-one laws and the minimum of a Markov process


Author: P. W. Millar
Journal: Trans. Amer. Math. Soc. 226 (1977), 365-391
MSC: Primary 60J25; Secondary 60F20, 60J30, 60J55
DOI: https://doi.org/10.1090/S0002-9947-1977-0433606-6
MathSciNet review: 0433606
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Abstract: If $ \{ {X_t},t > 0\} $ is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on I, the value of this minimum, the process $ \{ X(M + t),t > 0\} $ is shown to be Markov with stationary transitions which depend on I. For a wide class of Markov processes, including those obtained from Lévy processes via time change and multiplicative functional, a zero-one law is shown to hold at M in the sense that $ { \cap _{t > 0}}\sigma \{ X(M + s),s \leqslant t\} = \sigma \{ X(M)\} $, modulo null sets. When such a law holds, the evolution of $ \{ X(M + t),t \geqslant 0\} $ depends on events before M only through $ X(M)$ and I.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0433606-6
Keywords: Zero-one laws, minimum, strong Markov process, stationary independent increments, random time, last exit time, path decomposition, h path transform
Article copyright: © Copyright 1977 American Mathematical Society

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