By Daniel I.A. Cohen
ISBN-10: 0471035351
ISBN-13: 9780471035350
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V. a)c. s. e a. e. defined on n0• n: f f However, if we take I (Chapter II, Propo- The following random variables are a T(t) = t + Ta o LI Vt It will be convenient to define these random variables everywhere, that is, to define T(dcu), T(dcu) if Xt(cu) = oo. Since S00(cu) has empty interior, we can define for instance We know that the post-T(t) processes are Markov chains: their entrance laws will play a basic role in the sequel. PROPOSITION 1. Let a be an element of E. There is one and only one normalized entrance law (~j(t )) relative to II such that, for every initial lawµ.
S. s. if the parameter set of X is To). Then the entrance law of the post-T process BOUNDARY THEORY FOR MARKOV CHAINS 26 is given by: PIY(t) = il = I PIX(T) = j, T . i(t) J (i € I), = PIT = ool ; (thus the value t = 0 can be added to the parameter set of the post-T process). Moreover, the a -field I;! T• M € ~T' i €I, PILnM\X(T)= il = PIL\X(T)= i}PIMjX(T)= i} . , let tv, v = 1, ... , n be strictly positive numbers in increasing order, jv, v = 1, ... , n be states in I. If we discard a set of measure 0, we may assume that X(t) € I for every rational t.
If w (or X(t, w)) = w (t), and if w € n0 € n, we set Xt(w) we denote the limit at 0 by X 0 (w) (or X(O, w)). We also define Xt_(w) (or X(t-, w)) = w (t-). v. 's X(t), t > 0) such that X(t) is a Markov process with the transition semi-group Il ( ·) and the entrance law p (. ). p is concentrated on no iff p(t) = µII (t) for some probability law µ. on I , and then µ. is just the law of X(O). We shall not use a notation emphasizing the dependence of P on p ( · ), except in a few cases, the most noteworthy of which are: 1) If p(t) = µII (t) for some law µ.
Basic techniques of combinatorial theory by Daniel I.A. Cohen
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