scholarly journals A functional CLT for nonconventional polynomial arrays

2020 ◽  
Vol 40 (5) ◽  
pp. 2827-2873
Author(s):  
Yeor Hafouta ◽  
Keyword(s):  
Functional Clt

Functional Central Limit Theorem for Empirical Processes

2019 ◽  
pp. 428-437
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Empirical Process ◽  
Gaussian Approximation ◽  
Type Inequality ◽  
Stationary Sequence ◽  
General Criterion ◽  
Functional Clt ◽  
Finite Dimensional ◽  
Rosenthal Type Inequality ◽  
Mixing Sequences ◽  
Functional Central Limit

Here we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ‎-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α‎-dependent sequences whose α‎-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β‎-dependent sequences are also investigated by directly proving tightness of the empirical process.

On the invariance principle for reversible Markov chains

2016 ◽  
Vol 53 (2) ◽  
pp. 593-599 ◽  
Author(s):  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Stochastic Processes ◽  
Partial Sums ◽  
Additive Functionals ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

Abstract In this paper we investigate the functional central limit theorem (CLT) for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional CLT is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.

scholarly journals Functional CLT for Random Walk Among Bounded Random Conductances

2007 ◽  
Vol 12 (0) ◽  
pp. 1323-1348 ◽  
Author(s):  
Marek Biskup ◽  
Timothy Prescott
Keyword(s):  
Random Walk ◽  
Functional Clt ◽  
Random Conductances

scholarly journals On the Functional Central Limit Theorem for Reversible Markov Chains with Nonlinear Growth of the Variance

2012 ◽  
Vol 49 (4) ◽  
pp. 1091-1105 ◽  
Author(s):  
Martial Longla ◽  
Costel Peligrad ◽  
Magda Peligrad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Central Limit ◽  
Adjoint Operator ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Conditional Convergence ◽  
Functional Clt ◽  
Functional Central Limit

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

scholarly journals Functional CLT for sample covariance matrices

Bernoulli ◽  
2010 ◽  
Vol 16 (4) ◽  
pp. 1086-1113 ◽  
Author(s):  
Zhidong Bai ◽  
Xiaoying Wang ◽  
Wang Zhou
Keyword(s):  
Covariance Matrices ◽  
Sample Covariance Matrices ◽  
Functional Clt ◽  
Sample Covariance

On the Functional Central Limit Theorem for Reversible Markov Chains with Nonlinear Growth of the Variance

2012 ◽  
Vol 49 (04) ◽  
pp. 1091-1105
Author(s):  
Martial Longla ◽  
Costel Peligrad ◽  
Magda Peligrad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Markov Chains ◽  
Central Limit ◽  
Adjoint Operator ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Conditional Convergence ◽  
Functional Clt ◽  
Functional Central Limit

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

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