Kolmogorov's and Mourier's Strong Laws for Arrays with Independent and Identically Distributed Columns

1984 ◽  
Author(s):  
Arif Zaman ◽  
Asad Zaman
Keyword(s):  
Strong Laws ◽  
Independent And Identically Distributed

scholarly journals On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Aiting Shen ◽  
Ying Zhang ◽  
Andrei Volodin
Keyword(s):  
Strong Convergence ◽  
Mild Condition ◽  
Complete Convergence ◽  
Random Variables ◽  
Dependent Random Variables ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Independent And Identically Distributed

Letan,n≥1be a sequence of positive constants withan/n↑and letX,Xn,n≥1be a sequence of pairwise negatively quadrant dependent random variables. The complete convergence for pairwise negatively quadrant dependent random variables is studied under mild condition. In addition, the strong laws of large numbers for identically distributed pairwise negatively quadrant dependent random variables are established, which are equivalent to the mild condition∑n=1∞PX>an<∞. Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables.

scholarly journals The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽  
2021 ◽  
Author(s):  
Krzysztof Jasiński
Keyword(s):  
Random Variables ◽  
Continuous Case ◽  
Coherent System ◽  
Coherent Systems ◽  
Discrete Random Variables ◽  
Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

A point-based Bayesian hierarchical model to predict the outcome of tennis matches

2019 ◽  
Vol 15 (4) ◽  
pp. 313-325 ◽  
Author(s):  
Martin Ingram
Keyword(s):  
Random Walk ◽  
Hierarchical Model ◽  
Bayesian Hierarchical Model ◽  
Bayesian Hierarchical ◽  
Gaussian Random Walk ◽  
Over Time ◽  
Modelling Approach ◽  
Independent And Identically Distributed

Abstract A well-established assumption in tennis is that point outcomes on each player’s serve in a match are independent and identically distributed (iid). With this assumption, it is enough to specify the serve probabilities for both players to derive a wide variety of event distributions, such as the expected winner and number of sets, and number of games. However, models using this assumption, which we will refer to as “point-based”, have typically performed worse than other models in the literature at predicting the match winner. This paper presents a point-based Bayesian hierarchical model for predicting the outcome of tennis matches. The model predicts the probability of winning a point on serve given surface, tournament and match date. Each player is given a serve and return skill which is assumed to follow a Gaussian random walk over time. In addition, each player’s skill varies by surface, and tournaments are given tournament-specific intercepts. When evaluated on the ATP’s 2014 season, the model outperforms other point-based models, predicting match outcomes with greater accuracy (68.8% vs. 66.3%) and lower log loss (0.592 vs. 0.641). The results are competitive with approaches modelling the match outcome directly, demonstrating the forecasting potential of the point-based modelling approach.

scholarly journals A new proof for the generalized law of large numbers under Choquet expectation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jing Chen ◽  
Zengjing Chen
Keyword(s):  
Probability Theory ◽  
Law Of Large Numbers ◽  
Moment Condition ◽  
Large Numbers ◽  
Choquet Expectation ◽  
The Law ◽  
Linear Probability ◽  
First Moment ◽  
Independent And Identically Distributed

Abstract In this article, we employ the elementary inequalities arising from the sub-linearity of Choquet expectation to give a new proof for the generalized law of large numbers under Choquet expectations induced by 2-alternating capacities with mild assumptions. This generalizes the Linderberg–Feller methodology for linear probability theory to Choquet expectation framework and extends the law of large numbers under Choquet expectation from the strong independent and identically distributed (iid) assumptions to the convolutional independence combined with the strengthened first moment condition.

scholarly journals Signature-Based Representations for the Reliability of Systems with Heterogeneous Components

2011 ◽  
Vol 48 (03) ◽  
pp. 856-867 ◽  
Author(s):  
Jorge Navarro ◽  
Francisco J. Samaniego ◽  
N. Balakrishnan
Keyword(s):  
Coherent Systems ◽  
Representation Theorems ◽  
Performance Of Systems ◽  
Independent And Identically Distributed

Signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes have proven very useful in studying the ageing characteristics of such systems and in comparing the performance of different systems under varied criteria. In this paper we consider extensions of these results to systems with heterogeneous components. New representation theorems are established for both the case of components with independent lifetimes and the case of component lifetimes under specific forms of dependence. These representations may be used to compare the performance of systems with homogeneous and heterogeneous components.

Strong laws for the maximal k-spacing when k?c log n

1984 ◽  
Vol 66 (3) ◽  
pp. 315-334 ◽  
Author(s):  
Paul Deheuvels ◽  
Luc Devroye
Keyword(s):  
Strong Laws

Algorithms and Formulae for Conversion Between System Signatures and Reliability Functions

2015 ◽  
Vol 52 (02) ◽  
pp. 490-507
Author(s):  
Jean-Luc Marichal
Keyword(s):  
System Reliability ◽  
Reliability Function ◽  
Efficient Algorithms ◽  
System Designs ◽  
Independent And Identically Distributed

The concept of a signature is a useful tool in the analysis of semicoherent systems with continuous, and independent and identically distributed component lifetimes, especially for the comparison of different system designs and the computation of the system reliability. For such systems, we provide conversion formulae between the signature and the reliability function through the corresponding vector of dominations and we derive efficient algorithms for the computation of any of these concepts from any other. We also show how the signature can be easily computed from the reliability function via basic manipulations such as differentiation, coefficient extraction, and integration.

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