On weak laws of large numbers

1970 ◽  
Vol 71 (6) ◽  
pp. 266-274 ◽  
Author(s):  
Z. Govindarajulu
Keyword(s):  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Weak Laws

scholarly journals Limit theorems for multivariate Brownian semistationary processes and feasible results

2019 ◽  
Vol 51 (03) ◽  
pp. 667-716
Author(s):  
Riccardo Passeggeri ◽  
Almut E. D. Veraart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Laws Of Large Numbers ◽  
Stationary Increments ◽  
Large Numbers ◽  
Weak Laws

AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

scholarly journals Moderate Laws of Large Numbers via Weak Laws

2020 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Martingale Theory ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Weak Laws

By a $moderate$ $law$ $of$ $large$ $numbers$ we mean any theorem whose conclusion includes the $L^{p}$-vanishment of the sequence of the sample means of some centered random variables with $1 \leq p < +\infty$ given.Given any $1 \leq p < +\infty$ and any $\eps > 0$,we prove a moderate law of large numbers for $L^{p+\eps}$-bounded random variables that obey a weak law.Thus our moderate laws in particular complement those obtained from the martingale theory,and establish the counterintuitive fact that (for$L^{p+\eps}$-bounded random variables) where there is a weak law there is a moderate law.

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