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Unformatted text preview: Poisson Processes Class Notes for ISEN 618 September 27, 2007 1 Definitions and Preliminaries 1.1 Stochastic Processes Definition 1 A realvalued stochastic process is a family of random variables X = { X ( t ) ,t ∈ T } indexed by a set T , all defined on the same probability space (Ω , F ,P ) . For our purposes, we will typically take the set T to be either R + = [0 , ∞ ) or N + = { , 1 , 2 ,... } . The random functions X ( · ,ω ) : T → R , ω ∈ Ω are called sample paths or trajectories . A stochastic process will be described as, for example, rightcontinuous, increasing, nondecreasing, of bounded variation, etc., if the sample paths have these properties (a.s.). A stochastic process is said to be integrable if sup ≤ t< ∞ E  X ( t )  < ∞ ; square integrable if sup ≤ t< ∞ EX ( t ) 2 < ∞ ; bounded if there exists a finite constant Γ such that P ( sup ≤ t< ∞  X ( t )  < Γ) = 1 . 1 1.2 Filtrations Definition 2 A family of sub σalgebras {F t ,t ≥ } is called increasing if s ≤ t implies F s ∈ F t . An increasing family of sub σalgebras is called a filtration ; we will typically denote a filtration by F . When F = {F t ,t ≥ } is a filtration, the σalgebra T h> F t + h is denoted F t + . The corresponding limit from the left, F t , is the smallest σalgebra containing all the sets in S h> F t h and is written W h> F t h . A filtration is right continuous if, for any t , F t + = F t . The most natural filtration for a stochastic process X is called the “history”; i.e. F t = σ ( X ( s ) , ≤ s ≤ t ), the smallest σalgebra with respect to which each of the variables X ( s ) , ≤ s ≤ t is measurable. A stochastic process { X ( t ) ,t ≥ } is adapted to a filtration {F t ,t ≥ } if for every t ≥ ,X ( t ) is F tmeasurable. 1.3 Point Processes and Counting Processes A point process is a random distribution of points in a Euclidean space. For example, the points may represent shocks that occur randomly over time and that cause damage to a system (here, if we start observing the shocks at time 0, the space is R + ). Or, the points may represent the location of stars in a galaxy (here the space is R 3 ), or the locations of factories that may produce airborne pollutants, or the locations of defects on a semiconductor wafer (here the space is R 2 ). For ease of exposition, in this discussion, we’ll consider point processes on R + , and we will think of the points as times of occurence of a particular phenomenon. Let us begin as usual with a probability space (Ω , F ,P ). We have the fol lowing formal definition of a point process. Definition 3 A point process T = { T n ,n ∈ N + } is an ordered sequence of nonnegative random variables 0 = T < T 1 < ··· denoting the locations of 2 points....
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 Fall '07
 Klutke

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