scholarly journals Processes with Block-Associated Increments

2011 ◽  
Vol 48 (2) ◽  
pp. 514-526 ◽  
Author(s):  
Adam Jakubowski ◽  
Joanna Karłowska-Pik
Keyword(s):  
Random Variables ◽  
Positive Association ◽  
Interpolation Formula ◽  
Simple Relationship ◽  
Random Vectors ◽  
Covariance Functions ◽  
The Central Limit Theorem ◽  
Weak Association ◽  
Independence Of Random Variables ◽  
Vector Valued

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

scholarly journals Processes with Block-Associated Increments

2011 ◽  
Vol 48 (02) ◽  
pp. 514-526
Author(s):  
Adam Jakubowski ◽  
Joanna Karłowska-Pik
Keyword(s):  
Random Variables ◽  
Positive Association ◽  
Interpolation Formula ◽  
Simple Relationship ◽  
Random Vectors ◽  
Covariance Functions ◽  
The Central Limit Theorem ◽  
Weak Association ◽  
Independence Of Random Variables ◽  
Vector Valued

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

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.

MONOTONIC INDEPENDENCE ON THE WEAKLY MONOTONE FOCK SPACE AND RELATED POISSON TYPE THEOREM

2005 ◽  
Vol 08 (02) ◽  
pp. 259-275 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Direct Proof ◽  
Binomial Coefficients ◽  
Second Quantization ◽  
The Central Limit Theorem ◽  
Natural Way ◽  
Poisson Type

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

scholarly journals On the estimate of the remainder term in the central limit theorem for non-equally distributed random variables

1972 ◽  
Vol 12 (4) ◽  
pp. 183-194
Author(s):  
V. Paulauskas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Random Variables ◽  
The Central Limit Theorem

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. Паулаускас. Оценка скорости сходимости в центральной предельной теореме для разнораспределенных слагаемых V. Paulauskas. Konvergavimo greičio įvertinimas centrinėje ribinėje teoremoje nevienodai pasiskirsčiusiems dėmenims

scholarly journals Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mingzhou Xu ◽  
Kun Cheng
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Uniform Convergence ◽  
Central Limit ◽  
Random Variables ◽  
Precise Asymptotics ◽  
Partial Sum ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.

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