The branching diffusion with immigration

1980 ◽  
Vol 17 (1) ◽  
pp. 1-15 ◽  
Author(s):  
B. G. Ivanoff
Keyword(s):  
Steady State ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Measure ◽  
Steady State Distribution ◽  
Mixing Processes ◽  
Renormalization Transformation ◽  
Branching Diffusion ◽  
And Diffusion

The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.

The branching diffusion with immigration

1980 ◽  
Vol 17 (01) ◽  
pp. 1-15 ◽  
Author(s):  
B. G. Ivanoff
Keyword(s):  
Steady State ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Measure ◽  
Steady State Distribution ◽  
Mixing Processes ◽  
Renormalization Transformation ◽  
Branching Diffusion ◽  
And Diffusion

The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.

A Central Limit Theorem for Globally Nonstationary Near-Epoch Dependent Functions of Mixing Processes

1992 ◽  
Vol 8 (3) ◽  
pp. 313-329 ◽  
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Stochastic Processes ◽  
Central Limit ◽  
Mixing Processes ◽  
Martingale Differences

A central limit theorem is proved for dependent stochastic processes. Global heterogeneity of the distribution of the terms is permitted, including asymptotically unbounded moments. The approach is to adapt a CLT for martingale differences due to McLeish and show that suitably defined Bernstein blocks satisfy the required conditions.

FCLTs for Dependent Variables

2021 ◽  
pp. 699-723
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Time Domain ◽  
Functional Central Limit Theorem ◽  
Mixing Processes ◽  
Brownian Motions ◽  
Dependent Variables ◽  
The Time Domain ◽  
Functional Central Limit

After some technical preliminaries, this chapter gives two contrasting proofs of the functional central limit theorem for near‐epoch dependent functions of mixing processes. It goes on to consider variants of the result for nonstationary increments in which the limits are transformed Brownian motions, subject to distortions of the time domain. The multivariate case of the result is also given.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS I

2000 ◽  
Vol 16 (5) ◽  
pp. 621-642 ◽  
Author(s):  
Robert M. de Jong ◽  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Unit Root ◽  
Functional Central Limit Theorem ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Unit Root Testing ◽  
Functional Central Limit

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

A Martingale Approach to Regenerative Simulation

1995 ◽  
Vol 9 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peter W. Glynn ◽  
Donald L. Iglehart
Keyword(s):  
Steady State ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Martingale Approach ◽  
Regenerative Simulation

The standard regenerative method for estimating steady-state parameters is extended to permit cycles that begin and end in different states. This result is established using the Dynkin martingale and a related solution to Poisson's equation. We compare the variance constant that appears in the associated central limit theorem with that arising from cycles that begin and end in the same state. The standard regenerative method has a smaller variance constant than does the alternative.

CLTs for Dependent Processes

2021 ◽  
pp. 548-567
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Dependent Processes

This chapter deals with the central limit theorem (CLT) for dependent processes. As with the law of large numbers, the focus is on near‐epoch dependent and mixing processes and array versions of the results are given to allow heterogeneity. The cornerstone of these results is a general CLT due to McLeish, from which a result for martingales is obtained directly. A result for stationary ergodic mixingales is given, and the rest of the chapter is devoted to proving and interpreting a CLT for mixingales and hence for arrays that are near‐epoch dependent on a strong‐mixing and uniform-mixing processes.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS II

2000 ◽  
Vol 16 (5) ◽  
pp. 643-666 ◽  
Author(s):  
James Davidson ◽  
Robert M. de Jong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Convergence Results ◽  
Functional Central Limit

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

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