Approximation of population processes by Thomas G. Kurtz

By Thomas G. Kurtz

Inhabitants tactics are stochastic versions for structures regarding a couple of comparable debris. Examples comprise types for chemical reactions and for epidemics. The version could contain a finite variety of attributes, or perhaps a continuum.

This monograph considers approximations which are attainable while the variety of debris is big. The versions thought of will contain a finite variety of varieties of debris.

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Take η such that η = F −1 (ξ) holds (if there are multiple solutions, pick one arbitrarily). Then η will have the desired distribution. To see this, consider the following example. Let η be a random variable with   α1 with probability p1 η = α2 with probability p2  α with probability p , 3 3 where 3 i=1 pi = 1 and pi ≥ 0   α1 η = α2  α 3 for i = 1, 2, 3. Then F (η) = ξ implies if ξ∈[0, p1 ] if ξ∈(p1 , p1 + p2 ] if ξ∈(p1 + p2 , 1]. 3. MONTE CARLO METHODS 29 This can be generalized to any countable number of discrete values in the range of η, and since any function can be approximated by a step function, the results hold for any probability distribution function F .

For simplicity, we assume that η1 and η2 have densities with mean zero. Then Var[η1 + η2 ] = E[(η1 + η2 − E[η1 + η2 ])2 ] = E[(η1 + η2 )2 ] = (x + y)2 fη1 η2 (x, y) dx dy = x2 fη1 η2 (x, y) dx dy + +2 y 2 fη1 η2 (x, y) dx dy xyfη1 η2 (x, y) dx dy. The first two integrals are equal to Var(η1 ) and Var(η2 ), respectively. The third integral is zero. Indeed, because η1 and η2 are independent, fη1 η2 (x, y) = fη1 (x)fη2 (y) and xyfη1 η2 (x, y) dx dy = xfη1 (x) dx yfη2 (y) dy = E[η1 ]E[η2 ] = 0. 2. EXPECTED VALUES AND MOMENTS 27 Another simple property of the variance is that Var(aη) = a2 Var(η), where a is a constant.

Let η1 and η2 be random variables and define the inner product by (η1 , η2 ) = E[η1 η2 ]. Since E [(η − E[η|ξ])h(ξ)] vanishes for all h(ξ), we see that η − E[η|ξ] is perpendicular to all functions h(ξ). Set P η = E[η|ξ]. Then η = P η + (η − P η) with (η − P η, P η) = 0, and we can interpret P η as the orthogonal projection of η onto the subspace of random variables that are functions of ξ and have finite variance. 7. BAYES’ THEOREM 41 We now consider the special case where η and ξ are random variables whose joint density fηξ is known: P (s < η ≤ s + ds, t < ξ ≤ t + dt) = fηξ (s, t) ds dt.

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