scholarly journals A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees

2016 ◽  
Vol 26 (6) ◽  
pp. 3659-3698 ◽  
Author(s):  
Rudolf Grübel ◽  
Zakhar Kabluchko
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Random Walks ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Random Trees ◽  
Branching Random Walks ◽  
Functional Central Limit

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.

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.

scholarly journals A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees

2018 ◽  
Vol 55 (2) ◽  
pp. 610-626 ◽  
Author(s):  
Adam Bowditch
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Upper Bound ◽  
Functional Central Limit Theorem ◽  
Biased Random Walk ◽  
Quenched Central Limit Theorem ◽  
Functional Central Limit

AbstractIn this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

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