On the functional central limit theorem and the law of the iterated logarithm for Markov processes

1982 ◽  
Vol 60 (2) ◽  
pp. 185-201 ◽  
Author(s):  
R. N. Bhattacharya
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Markov Processes ◽  
Functional Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law ◽  
Functional Central Limit

scholarly journals A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

2016 ◽  
Vol 53 (4) ◽  
pp. 1178-1192 ◽  
Author(s):  
Alexander Iksanov ◽  
Zakhar Kabluchko
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Branching Random Walk ◽  
Iterated Logarithm ◽  
Functional Central Limit

Abstract Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W∞(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W∞(θ)-Wn(θ) as n→∞.

scholarly journals Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mingzhou Xu ◽  
Kun Cheng
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Uniform Convergence ◽  
Central Limit ◽  
Random Variables ◽  
Precise Asymptotics ◽  
Partial Sum ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.

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