scholarly journals A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

2011 ◽  
Vol 48 (02) ◽  
pp. 527-546 ◽  
Author(s):  
Patrizia Berti ◽  
Irene Crimaldi ◽  
Luca Pratelli ◽  
Pietro Rigo

Let X n be a sequence of integrable real random variables, adapted to a filtration (G n ). Define C n = √{(1 / n)∑ k=1 n X k − E(X n+1 | G n )} and D n = √n{E(X n+1 | G n ) − Z}, where Z is the almost-sure limit of E(X n+1 | G n ) (assumed to exist). Conditions for (C n , D n ) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑ k=1 n X_k - Z} = C n + D n → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

2011 ◽  
Vol 48 (2) ◽  
pp. 527-546 ◽  
Author(s):  
Patrizia Berti ◽  
Irene Crimaldi ◽  
Luca Pratelli ◽  
Pietro Rigo

Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk − E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) − Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.


2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.


2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).


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