Weak Convergence

2021 ◽  
pp. 638-667
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Representation Theorem ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Probability Measures ◽  
Functional Central Limit ◽  
Spaces Of Measures

This chapter reviews the theory of weak convergence in metric spaces. Topics include Skorokhod’s representation theorem, the metrization of spaces of measures, and the concept of tightness of probability measures. The key relation is shown between weak convergence and uniform tightness. Considering the space C of continuous functions in particular, the functional central limit theorem is proved for martingales, together with extensions to the multivariate case.

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

2000 ◽  
Vol 16 (5) ◽  
pp. 621-642 ◽  
Author(s):  
Robert M. de Jong ◽  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Unit Root ◽  
Functional Central Limit Theorem ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Unit Root Testing ◽  
Functional Central Limit

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

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.

scholarly journals Central limit theorem for alternating renewal processes

2007 ◽  
Vol 47 ◽  
Author(s):  
Rimas Banys
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Regularity Condition ◽  
Distribution Functions ◽  
Renewal Processes ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

Functional central limit theorems for stationary alternating renewal processes with dependent work and repair times, and for associated workload processes are stated. The weak convergence of distributions of properly scaled processesin the Skorokhodspace holds under some regularity condition imposed on the distribution functions of work and repair times.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

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.

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