This book provides a pedagogical examination of the way in which stochastic models are encountered in applied sciences and techniques such as physics, engineering, biology and genetics, economics and social sciences. It covers Markov and semi-Markov models, as well as their particular cases: Poisson, renewal processes, branching processes, Ehrenfest models, genetic models, optimal stopping, reliability, reservoir theory, storage models, and queuing systems. Given this comprehensive treatment of the subject, students and researchers in applied sciences, as well as anyone looking for an introduction to stochastic models, will find this title of invaluable use.
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Yes, you can access Introduction to Stochastic Models by Marius Iosifescu,Nikolaos Limnios,Gheorghe Oprisan in PDF and/or ePUB format, as well as other popular books in Mathematics & Probability & Statistics. We have over one million books available in our catalogue for you to explore.
In this introductory chapter we present some notions regarding sequences of random variables (r.v.) and the most important classes of random processes.
1.1. Sequences of random variables
Let
be a probability space and (An) a sequence of events in
.
– The set of ω ∈ Ω belonging to an infinity of An, that is,
is called the upper limit of the sequence (An).
– The set of ω ∈ Ω belonging to all An except possibly to a finite number of them, that is,
is called the lower limit of the sequence (An).
THEOREM 1.1.– (Borel-Cantelli lemma)Let(An)be a sequence of events in
.
1.
, them
2.
and(An)is a sequence of independent events, then
Let (Xn) be a sequence of r.v. and X an r.v., all of them defined on the same probability space
. The convergence of the sequence (Xn) to X will be defined as follows:
1. Almost sure
2. In probability
if for any ε > 0,
.
3. In distribution(or weekly, or in law)
pointwise in every continuity point of FX, where FXn and FX are the distribution functions of Xn and X, respectively.
4. In mean of order
for all n ∈ N+, and if
. The most commonly used are the cases p = 1 (convergence in mean) and p = 2 (mean square convergence).
The relations between these types of convergence are as follows:
[1.1]
The convergence in distribution of r.v. is a convergence property of their distributions (i.e. of their laws) and it is the most used in probability applications. This con...
