scholarly journals A note on convergence results for varying interval valued multisubmeasures

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Anca Croitoru ◽  
Alina GavriluŢ ◽  
Alina Iosif ◽  
Anna Rita Sambucini
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Convergence Results ◽  
Interval Valued

<p style='text-indent:20px;'>Some limit theorems are presented for Riemann-Lebesgue integrals where the functions <inline-formula><tex-math id="M1">\begin{document}$ G_n $\end{document}</tex-math></inline-formula> and the measures <inline-formula><tex-math id="M2">\begin{document}$ M_n $\end{document}</tex-math></inline-formula> are interval valued and the convergence for the multisubmeasures is setwise. In particular sufficient conditions in order to obtain <inline-formula><tex-math id="M3">\begin{document}$ \int G_n dM_n \to \int G dM $\end{document}</tex-math></inline-formula> are given.</p>

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

Local limit theorems for generalized scheme of allocation of particles into ordered cells

2021 ◽  
Vol 31 (4) ◽  
pp. 293-307
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Weak Convergence ◽  
Negative Binomial Distribution ◽  
Limit Theorems ◽  
Negative Binomial ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Number Of Components ◽  
Normal Law ◽  
Local Limit Theorems ◽  
Generalized Scheme

Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

1992 ◽  
Vol 24 (02) ◽  
pp. 241-266
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Optimal Stopping ◽  
Limit Theorems ◽  
Continuous Mapping ◽  
Random Variables ◽  
Exit Times ◽  
Convergence Results ◽  
Process Convergence ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Linear Cost

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (03) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ &lt; 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ &lt; 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

scholarly journals Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 749 ◽  
Author(s):  
Yury Khokhlov ◽  
Victor Korolev ◽  
Alexander Zeifman
Keyword(s):  
Limit Theorems ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Random Vectors ◽  
Scale Mixtures ◽  
Elliptically Contoured ◽  
Necessary And Sufficient

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Complements to heavy traffic limit theorems for the GI/G/1 queue

1972 ◽  
Vol 9 (1) ◽  
pp. 185-191 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Convergence Of Moments

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

scholarly journals Measure change in multitype branching

2004 ◽  
Vol 36 (2) ◽  
pp. 544-581 ◽  
Author(s):  
J. D. Biggins ◽  
A. E. Kyprianou
Keyword(s):  
Branching Processes ◽  
Sufficient Conditions ◽  
Branching Random Walk ◽  
Stochastic Domination ◽  
Moment Conditions ◽  
Branching Brownian Motion ◽  
Mean Convergence ◽  
Convergence Results ◽  
Watson Process ◽  
The One

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

scholarly journals Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

2015 ◽  
Vol 30 ◽  
pp. 843-870 ◽  
Author(s):  
Cheng-yi Zhang ◽  
Dan Ye ◽  
Cong-Lei Zhong ◽  
SHUANGHUA SHUANGHUA
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Numerical Linear Algebra ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Sufficient Conditions For Convergence ◽  
Diagonally Dominant ◽  
Necessary And Sufficient ◽  
Theoretical Results

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn&gt;0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

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