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 techniques in real-life problems.

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

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

The recommendations are basic for plenty of functions, yet they don't seem to be as completely awarded in different books at the topic as they're here.

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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. , but there are supplementary properties which are very important in stochastic modelling.

Also, we define a new random variable A'^ giving the total number of renewals on [0,L). >0}. 19) 50 Chapter 2 In reliability theory, the event {N = k} would mean that the (A:+l)th component introduced in the system would have an infinite lifetime! 21) and in general, for A: G N : P{N = k) = (F(+oo))' (1 -F(+oo)). , A^ = +oo. 26): So, it is possible to compute the mean of the total number of renewals very easily in the transient case. We can also give the distribution function of L. Indeed, we have: P{L

S. ,(2;): E[Xp,){(D)= \x{co')p{dco\co). v. X, with values in the measurable space {E^ii/), and denoted by (J{X) . This means that we are only interested in the following conditional probabilities: P[Ap,),AscT{X). ) is 3 -measurable, (iii) for every event A belonging to (j(X) and for every event B belonging to 3 , we have: \C(A, co)P{dco) = P{A n B). ) from defined by: Q{S,(D) = p[{co': X(co') e 5|3, ))(^),5 e y/. y/xQ. 69) The problem of the existence of regular conditional probability was solved by Loeve (1977) or Gikhman and Skorokhod (1980).

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