scholarly journals Stable Limit Theorems for Empirical Processes under Conditional Neighborhood Dependence

2017 ◽  
Author(s):  
Ji Hyung Lee ◽  
Kyungchul Song
Keyword(s):  
Limit Theorems ◽  
Empirical Processes ◽  
Stable Limit

scholarly journals Stable limit theorems for empirical processes under conditional neighborhood dependence

Bernoulli ◽  
2019 ◽  
Vol 25 (2) ◽  
pp. 1189-1224 ◽  
Author(s):  
Ji Hyung Lee ◽  
Kyungchul Song
Keyword(s):  
Limit Theorems ◽  
Empirical Processes ◽  
Stable Limit

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

scholarly journals Some Limit Theorems for Empirical Processes: Discussion of the Paper of Professors Gine and Zinn

1984 ◽  
Vol 12 (4) ◽  
pp. 990-998
Author(s):  
Kenneth S. Alexander ◽  
R. M. Dudley ◽  
Peter Gaenssler ◽  
Walter Philipp ◽  
David Pollard ◽  
...  
Keyword(s):  
Limit Theorems ◽  
Empirical Processes

scholarly journals α-Stable limit theorems for sums of dependent random vectors

1989 ◽  
Vol 29 (2) ◽  
pp. 219-251 ◽  
Author(s):  
Adam Jakubowski ◽  
Maria Kobus
Keyword(s):  
Limit Theorems ◽  
Stable Limit ◽  
Random Vectors

New Ways to Prove Central Limit Theorems

1985 ◽  
Vol 1 (3) ◽  
pp. 295-313 ◽  
Author(s):  
David Pollard
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Maximum Likelihood Estimators ◽  
Empirical Processes ◽  
Central Limit Theorems ◽  
Criterion Function ◽  
Nonstandard Conditions

This paper describes some techniques for proving asymptotic normality of statistics defined by maximization of random criterion function. The techniques are based on a combination of recent results from the theory of empirical processes and a method of Huber for the study of maximum likelihood estimators under nonstandard conditions.

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