scholarly journals A Cramér Type Theorem for Weighted Random Variables

2002 ◽  
Vol 7 (0) ◽  
Author(s):  
Jamal Najim
Keyword(s):  
Type Theorem ◽  
Random Variables ◽  
Weighted Random Variables

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 A De Finetti-type theorem for nonexchangeable finite-valued random variables

2008 ◽  
Vol 347 (2) ◽  
pp. 407-415 ◽  
Author(s):  
Ricardo Vélez Ibarrola ◽  
Tomás Prieto-Rumeau
Keyword(s):  
Type Theorem ◽  
Random Variables ◽  
De Finetti

scholarly journals On Some Laws of Large Numbers for Uncertain Random Variables

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2258
Author(s):  
Piotr Nowak ◽  
Olgierd Hryniewicz
Keyword(s):  
Type Theorem ◽  
Random Variables ◽  
Continuous Functions ◽  
Chance Theory ◽  
Laws Of Large Numbers ◽  
Uncertain Variables ◽  
Large Numbers ◽  
Delayed Sums ◽  
Special Cases ◽  
Independent Identically Distributed

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

scholarly journals On distribution of a maximum of random variables

2020 ◽  
pp. 124-136
Author(s):  
Степан Алексеевич Рогонов ◽  
Илья Сергеевич Солдатенко
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Maximum Function ◽  
The Monte Carlo Method ◽  
Weighted Random Variables ◽  
Fuzzy Random

В работе исследуется способ идентификации методом Монте-Карло распределений нечетких случайных величин, в параметрическом задании которых присутствует функция максимума от взвешенных случайных величин. In this paper, we study a method for identifying by the Monte Carlo method distributions of fuzzy random variables, in the parametric setting of which there is a maximum function of weighted random variables.

SOME NEW RESULTS ON RÉNYI ENTROPY OF RESIDUAL LIFE AND INACTIVITY TIME

2011 ◽  
Vol 25 (2) ◽  
pp. 237-250 ◽  
Author(s):  
Xiaohu Li ◽  
Shuhong Zhang
Keyword(s):  
Order Statistics ◽  
Residual Life ◽  
Random Variables ◽  
Record Values ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Weighted Distributions ◽  
Inactivity Time ◽  
Weighted Random Variables ◽  
Monotonic Properties

This article deals with Rényi entropies for the residual life and the inactivity time. Monotonic properties of the entropy in order statistics, record values, and weighted distributions are investigated, and the comparison on weighted random variables is studied in terms of residual Rényi entropy as well.

Limit Distributions for Sums of Weighted Random Variables

1975 ◽  
Vol 18 (2) ◽  
pp. 291-293 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Random Variables ◽  
Limit Distributions ◽  
Partial Sum ◽  
Image Position ◽  
Weighted Random Variables

AbstractLet X1, X2, X3 … be i.i.d., Sn their nth partial sum with Sn = 0; Suppose that

scholarly journals On the necessary condition for Baum-Katz type theorem for non-identically distributed and negatively dependent random fields

2018 ◽  
Vol 72 (2) ◽  
pp. 1
Author(s):  
Zbigniew Łagodowski
Keyword(s):  
Random Walks ◽  
Random Field ◽  
Random Fields ◽  
Complete Convergence ◽  
Type Theorem ◽  
Random Variables ◽  
Necessary Condition ◽  
Dependent Random Variables ◽  
Negatively Dependent Random Variables ◽  
Convergence Results

Let  \(\{ X_{\bf n}, {\bf n}\in \mathbb{N}^d \}\) be a random field of negatively dependent  random variables.  The complete  convergence results for negatively dependent  random fields  are refined. To obtain the main theorem several lemmas  for convergence of families indexed by \(\mathbb{N}^d\)   have been proved. Auxiliary lemmas have wider application to study  the random walks on the lattice.

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