Download e-book for kindle: Applied Semi-Markov Processes by Jacques Janssen, Raimondo Manca

By Jacques Janssen, Raimondo Manca

Aims to provide to the reader the instruments essential to follow semi-Markov strategies in real-life problems.

The ebook is self-contained and, ranging from a low point of likelihood strategies, progressively brings the reader to a deep wisdom of semi-Markov processes.

Presents homogeneous and non-homogeneous semi-Markov methods, in addition to Markov and semi-Markov rewards processes.

The options are primary for lots of functions, yet they don't seem to be as completely offered in different books at the topic as they're here.

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69) The problem of the existence of regular conditional probability was solved by Loeve (1977) or Gikhman and Skorokhod (1980). For our goal, let us just say that this is the case if 3 is generated by a finite or countable family of random variables or if the space £" is a complete separable metric space. v. ,X,(^')

E. 1 If F(0) < 1, then, for all t, N(t) has moments of any order, In particular, this proposition implies that the renewal function is finite for all finite t. 4): i/(0-XFW(0. 9) In several cases, it is useful to consider the initial renewal and to define at time t the random variable N^t) as being the total number of renewals on [ 0 , / ] . 6): R(t) = f^F^"\t). 14) R(t) = Uo(t)-^Hit). The classification of a renewal process is based on three concepts: recurrence, transience ^nd periodicity. 1 (i) A renewal process (T^, n>l) is recurrent if X^ < oo for all n; otherwise it is called transient.

17) to the general case. Particular cases (i) 3 , is generated by one r,v. X. v. ,X^. ,X^(CD) = x^, for instance. ,X^(CO) = X^] . 32) a result often used in the sequel to evaluate the mean of a random variable using its conditional expectation with respect to some given event. v. 27) becomes: jp{Ap,(co))dP = P(Af]BlBe:5,. 14). 38) P[A\S,){o)) = P{A\coea. v. 27), we have: ^(7|3,) = ^(F). 39) is proved. v. 41) allow us to have a better understanding of the intuitive meaning of conditioning. Under independence assumptions, conditioning has absolutely no impact, for example, on the expectation or the probability, and on the contrary dependence implies that the results with or without conditioning will be different, this fact meaning that we can interpret conditioning as given additional information useful to get more precise results in the case of dependence of course.

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