Introduction to Stochastic Processes
When considering technical, economic, ecological, or other problems, in several cases the quantities \(\left \{{X}_{t},\;t \in \mathcal{T}\right \}\) being examined can be regarded as a collection of random variables. This collection describes the changes
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Introduction to Stochastic Processes
2.1 Stochastic Processes When considering technical, economic, ecological, or other problems, in several cases the quantities fXt ; t 2 T g being examined can be regarded as a collection of random variables. This collection describes the changes (usually in time and in space) of considered quantities. If the set T is a subset of the set of real numbers, then the set ft 2 T g can be interpreted as time and we can say that the random quantities Xt vary in time. In this case the collection of random variables fXt ; t 2 T g is called a stochastic process. In mathematical modeling of randomly varying quantities in time, one might rely on the highly developed theory of stochastic processes. Definition 2.1. Let T R. A stochastic process X is defined as a collection X D fXt ; t 2 T g of indexed random variables Xt , which are given on the same probability space .; A; P .//. Depending on the notational complexity of the parameter, we occasionally interchange the notation Xt with X.t/. It is clear that Xt D Xt .!/ is a function of two variables. For fixed t 2 T , Xt is a random variable, and for fixed ! 2 , Xt is a function of the variable t 2 T , which is called a sample path of the stochastic process. Depending on the set T , X is called a discrete-time stochastic process if the index set T consists of consecutive integers, for example, T D f0; 1; : : :g or T D f: : : ; 1; 0; 1; : : :g. Further, X is called a continuous-time stochastic process if T equals an interval of the real line, for example, T D Œa; b, T D Œ0; 1/ or T D .1; 1/. Note that in the case of discrete time, X is a sequence fXn ; n 2 T g of random variables, while it determines a random function in the continuous-time case. It should be noted that similarly to the notion of real-valued stochastic processes, we may define complex or vector valued stochastic processes also if Xt take values in a complex plane or in higher-dimensional Euclidean space. L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, DOI 10.1007/978-1-4614-5317-8 2, © Springer Science+Business Media, LLC 2013
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2 Introduction to Stochastic Processes
2.2 Finite-Dimensional Distributions of Stochastic Processes A stochastic process fXt ; t 2 T g can be characterized in a statistical sense by its finite-dimensional distributions. Definition 2.2. The finite-dimensional distributions of a stochastic process fXt ; t 2 T g are defined by the family of all joint distribution functions Ft1 ;:::;tn .x1 ; : : : ; xn / D P .Xt1 < x1 ; : : : ; Xtn < xn /; where n D 1; 2; : : : and t1 ; : : : ; tn 2 T . The family of introduced distribution functions F D fFt1 ;:::;tn ; t1 ; : : : ; tn 2 T ; n D 1; 2; : : :g satisfies the following, specified consistency conditions: (a) For all positive integers n, m and indices t1 ; : : : ; tnCm 2 T lim
xnC1 !1
:::
lim
xnCm !1
Ft1 ;:::;tn ;tnC1 ;:::;tnCm .x1 ; : : : ; xn ; xnC1 ; : : : ; xnCm /
D Ft1 ;:::;tn .x1 ; : : : ; xn /; x1 ; : : : ; xn 2 R : (b) For all permutations .i1
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