Applied Semi-Markov Processes by Jacques Janssen, Raimondo Manca

By Jacques Janssen, Raimondo Manca

Aims to offer 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 strategies, progressively brings the reader to a deep wisdom of semi-Markov processes.

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

The techniques are basic 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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2) Let us suppose that r is a stopping time. We have: [co:T{CO)

V. 41) (2;r)2VdetL Then, it can be shown by integration that parameters |ii, 2 are indeed respectively the mean vector and the variance-covariance matrix of X As usual, we will use the notation: X -< A^„(fi,L). The characteristic function of Xis given by: ;>'t--t'z:t (p,{t) = e 2 . v. 43) X, +- + X^ -< N„iii, + - + Ji„,i:, + - + £ J . 1: ^2->"2^ ^i-/"i relations meaning that in this case, all the probability mass in the plan lies on a straight line so the two random variables X\^2 are perfectly dependent with probability 1.

25). 22) as a definition with the help of the Radon Nikodym theorem, Halmos (1974). 1 If 5, is a sub-a-algebra of 3 , the conditional expectation of the integrable r,v. v. of the equivalence class such that: (i) E^ (Y) is 3, -measurable, (ii) 1^3^ {Y){co)dP = \Y{co)dP,B e 3,. s. 27). 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.

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