The rate of convergence in the functional central limit theorem for random quadratic forms with some applications to the law of the iterated logarithm

1989 ◽  
Vol 107 (2) ◽  
pp. 137-153 ◽  
Author(s):  
T. Mikosch
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Quadratic Forms ◽  
Functional Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law ◽  
Functional Central Limit

scholarly journals A law of the iterated logarithm for martingales

1991 ◽  
Vol 43 (2) ◽  
pp. 181-185 ◽  
Author(s):  
Michael Voit
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Upper Bound ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Using a slight generalisation of Brown's inequality, we show that for martingales the existence of a weak nonuniform bound on the rate of convergence in the central limit theorem yields the usual upper bound part of the law of the iterated logarithm.

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→∞.

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