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 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.

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.

Reversible Markov Chains

2019 ◽  
pp. 405-427
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Gaussian Approximation ◽  
Building Blocks ◽  
Partial Sums ◽  
Reversible Markov Chain ◽  
Maximal Inequalities ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

THE IMPOSSIBILITY OF CONSISTENT DISCRIMINATION BETWEEN I(0) AND I(1) PROCESSES

2008 ◽  
Vol 24 (3) ◽  
pp. 616-630 ◽  
Author(s):  
Ulrich K. MÜller
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Wiener Process ◽  
Central Limit ◽  
Unit Root ◽  
Unit Root Tests ◽  
Partial Sums ◽  
Asymptotic Size ◽  
Stationarity Tests ◽  
Functional Central Limit

An I(0) process is commonly defined as a process that satisfies a functional central limit theorem, i.e., whose scaled partial sums converge weakly to a Wiener process, and an I(1) process as a process whose first differences are I(0). This paper establishes that with this definition, it is impossible to consistently discriminate between I(0) and I(1) processes. At the same time, on a more constructive note, there exist consistent unit root tests and also nontrivial inconsistent stationarity tests with correct asymptotic size.

P(𝜙)1-process for the spin-boson model and a functional central limit theorem for associated additive functionals

Stochastics ◽  
2017 ◽  
Vol 89 (6-7) ◽  
pp. 1104-1115 ◽  
Author(s):  
Soumaya Gheryani ◽  
Fumio Hiroshima ◽  
József Lőrinczi ◽  
Achref Majid ◽  
Habib Ouerdiane
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Additive Functionals ◽  
Boson Model ◽  
Functional Central Limit

scholarly journals A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM

2011 ◽  
Vol 28 (3) ◽  
pp. 671-679 ◽  
Author(s):  
Søren Johansen ◽  
Morten Ørregaard Nielsen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Fractional Integration ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Moment Condition ◽  
The Moment ◽  
Functional Central Limit ◽  
Fractional Functional

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of xt = Δ−dut, where $d\, \in \,\left({ - {1 \over 2}\,,\,{1 \over 2}} \right)$ is the fractional integration parameter and ut is weakly dependent. The classical condition is existence of q ≥ 2 and $q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $ moments of the innovation sequence. When d is close to $ - {1 \over 2}$ this moment condition is very strong. Our main result is to show that when $d\, \in \,\left({ - \,{1 \over 2},\,0} \right)$ and under some relatively weak conditions on ut, the existence of $q\, \ge \,\left({d\, + \,{1 \over 2}} \right)^{ - 1} $ moments is in fact necessary for the FCLT for fractionally integrated processes and that $q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $ moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) presented a fractional FCLT where only q > 2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient and hence that their result is incorrect.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS II

2000 ◽  
Vol 16 (5) ◽  
pp. 643-666 ◽  
Author(s):  
James Davidson ◽  
Robert M. de Jong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Convergence Results ◽  
Functional Central Limit

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

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