scholarly journals Convergence rates for weighted sums in noncommutative probability space

2014 ◽  
Vol 409 (2) ◽  
pp. 963-972 ◽  
Author(s):  
Byoung Jin Choi ◽  
Un Cig Ji
Keyword(s):  
Probability Space ◽  
Convergence Rates ◽  
Weighted Sums ◽  
Noncommutative Probability

Exponential convergence rates for weighted sums in noncommutative probability space

2016 ◽  
Vol 19 (04) ◽  
pp. 1650027 ◽  
Author(s):  
Byoung Jin Choi ◽  
Un Cig Ji
Keyword(s):  
Probability Space ◽  
Von Neumann Algebra ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Large Deviation ◽  
Type Inequality ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Noncommutative Probability ◽  
Von Neumann

We study exponential convergence rates for weighted sums of successive independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. Then we prove a noncommutative extension of the result for the exponential convergence rate by Baum, Katz and Read. As applications, we first study a large deviation type inequality for weighted sums in a noncommutative probability space, and secondly we study exponential convergence rates for weighted free additive convolution sums of probability measures.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

scholarly journals Some Types of Convergence for Negatively Dependent Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Ruixue Wang ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Complete Convergence ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Weighted Sums ◽  
Convergence Theorems ◽  
Dependent Random Variables ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables

In this paper, we research complete convergence and almost sure convergence under the sublinear expectations. As applications, we extend some complete and almost sure convergence theorems for weighted sums of negatively dependent random variables from the traditional probability space to the sublinear expectation space.

REFINEMENT OF CONVERGENCE RATES FOR WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007
Author(s):  
Si-Li Niu ◽  
Jong-Il Baek
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Counting Processes ◽  
Precise Asymptotics ◽  
Strong Law ◽  
Large Numbers ◽  
Passage Times

In this paper, we establish one general result on precise asymptotics of weighted sums for i.i.d. random variables. As a corollary, we have the results of Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368], Gut and Spătaru [Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000) 1870–1883; Precise asymptotics in the Baum–Katz and Davis laws of large numbers, J. Math. Anal. Appl. 248 (2000) 233–246], Gut and Steinebach [Convergence rates and precise asymptotics for renewal counting processes and some first passage times, Fields Inst. Comm. 44 (2004) 205–227] and Heyde [A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975) 173–175]. Meanwhile, we provide an answer for the possible conclusion pointed out by Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368].

scholarly journals Complete integration convergence for arrays of rowwise extended negatively dependent random variables under the sub-linear expectations

2021 ◽  
Vol 6 (11) ◽  
pp. 12166-12181
Author(s):  
Shuyan Li ◽  
◽  
Qunying Wu
Keyword(s):  
Limit Theorems ◽  
Probability Space ◽  
Random Variables ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Dependent Random Variables ◽  
Complete Integration ◽  
Moment Convergence ◽  
Negatively Dependent Random Variables

<abstract><p>Limit theorems of sub-linear expectations are challenging field that has attracted widespread attention in recent years. In this paper, we establish some results on complete integration convergence for weighted sums of arrays of rowwise extended negatively dependent random variables under sub-linear expectations. Our results generalize the complete moment convergence of the probability space to the sub-linear expectation space.</p></abstract>

scholarly journals Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

2013 ◽  
Vol 2013 ◽  
pp. 1-26 ◽  
Author(s):  
Shunli Hao
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
General Setting ◽  
Weighted Sums ◽  
Moment Condition ◽  
Martingale Differences ◽  
Large Numbers ◽  
The Law

We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

scholarly journals Complete Convergence for Weighted Sums of Widely Acceptable Random Variables under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Rong Hu ◽  
Qunying Wu
Keyword(s):  
Convergence Theorem ◽  
Probability Space ◽  
Complete Convergence ◽  
Choquet Integral ◽  
Random Variables ◽  
Weighted Sums ◽  
Sublinear Expectation ◽  
Acceptable Random Variables ◽  
Complete Convergence Theorem

Using different methods than the probability space, under the condition that the Choquet integral exists, we study the complete convergence theorem for weighted sums of widely acceptable random variables under sublinear expectation space. We proved corresponding theorem which was extended to the sublinear expectations’ space from the probability space, and similar results were obtained.

scholarly journals Double-ended queues and joint moments of left–right canonical operators on full Fock space

2015 ◽  
Vol 26 (02) ◽  
pp. 1550016 ◽  
Author(s):  
Mitja Mastnak ◽  
Alexandru Nica
Keyword(s):  
Simple Formula ◽  
Probability Space ◽  
Fock Space ◽  
Free Probability ◽  
Noncommutative Probability ◽  
Operator Model ◽  
Basic Operator ◽  
The Family ◽  
The Right ◽  
Free Independence

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (𝒜, φ), a family of "(ℓ, r )-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (ℓ, r )-cumulants of a (2d)-tuple (A1,…,Ad, B1,…,Bd), with A1,…,Ad canonical operators on the left and B1,…,Bd canonical operators on the right. This extends a known one-sided formula for free cumulants of A1,…,Ad, which establishes a basic operator model for the R-transform of free probability.

Convergence rates in the weak law of large numbers for weighted sums of i.i.d. random variables and applications in errors-in-variables models

2019 ◽  
Vol 19 (06) ◽  
pp. 1950041 ◽  
Author(s):  
Mingyang Zhang ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Regression Models ◽  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Weighted Sums ◽  
Errors In Variables ◽  
Error In Variables ◽  
Large Numbers ◽  
Necessary And Sufficient

We prove necessary and sufficient conditions for the convergence rates in the weak law of large numbers for weighted sums of independent and identically distributed (i.i.d.) random variables. This result is applied to simple linear error-in-variables regression models.

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