A limit theorem for discrete-parameter random evolutions

1984 ◽  
Vol 16 (2) ◽  
pp. 281-292
Author(s):  
Gilles A. Blum
Keyword(s):  
Limit Theorem ◽  
Difference Equations ◽  
Diffusion Approximations ◽  
Stochastic Difference Equations ◽  
Markovian Environment ◽  
Discrete Parameter ◽  
Fisher Model ◽  
Random Evolutions

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

A limit theorem for discrete-parameter random evolutions

1984 ◽  
Vol 16 (02) ◽  
pp. 281-292
Author(s):  
Gilles A. Blum
Keyword(s):  
Limit Theorem ◽  
Difference Equations ◽  
Diffusion Approximations ◽  
Stochastic Difference Equations ◽  
Markovian Environment ◽  
Discrete Parameter ◽  
Fisher Model ◽  
Random Evolutions

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

1977 ◽  
Vol 14 (1) ◽  
pp. 58-74 ◽  
Author(s):  
Harry A. Guess ◽  
John H. Gillespie
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Population Biology ◽  
Correction Term ◽  
Random Environments ◽  
Diffusion Approximations ◽  
Uhlenbeck Process ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Approach Zero

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

1977 ◽  
Vol 14 (01) ◽  
pp. 58-74 ◽  
Author(s):  
Harry A. Guess ◽  
John H. Gillespie
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Population Biology ◽  
Correction Term ◽  
Random Environments ◽  
Diffusion Approximations ◽  
Uhlenbeck Process ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Approach Zero

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

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