scholarly journals On the rates of convergence of Erlang's model

1999 ◽  
Vol 36 (4) ◽  
pp. 1167-1184 ◽  
Author(s):  
Christine Fricker ◽  
Philippe Robert ◽  
Danielle Tibi
Keyword(s):  
Rates Of Convergence ◽  
Upper Bounds ◽  
Probabilistic Methods ◽  
Convergence To Equilibrium ◽  
Coupling Techniques

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

scholarly journals On the rates of convergence of Erlang's model

1999 ◽  
Vol 36 (04) ◽  
pp. 1167-1184 ◽  
Author(s):  
Christine Fricker ◽  
Philippe Robert ◽  
Danielle Tibi
Keyword(s):  
Rates Of Convergence ◽  
Upper Bounds ◽  
Probabilistic Methods ◽  
Convergence To Equilibrium ◽  
Coupling Techniques

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

scholarly journals Minimax Rates of ℓp-Losses for High-Dimensional Linear Errors-in-Variables Models over ℓq-Balls

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 722
Author(s):  
Xin Li ◽  
Dongya Wu
Keyword(s):  
Additive Noise ◽  
Rates Of Convergence ◽  
Upper Bounds ◽  
High Dimensional ◽  
Regularity Conditions ◽  
Errors In Variables ◽  
Lower And Upper Bounds ◽  
Information Theoretic ◽  
Minimax Rates ◽  
Sparse Vector

In this paper, the high-dimensional linear regression model is considered, where the covariates are measured with additive noise. Different from most of the other methods, which are based on the assumption that the true covariates are fully obtained, results in this paper only require that the corrupted covariate matrix is observed. Then, by the application of information theory, the minimax rates of convergence for estimation are investigated in terms of the ℓp(1≤p<∞)-losses under the general sparsity assumption on the underlying regression parameter and some regularity conditions on the observed covariate matrix. The established lower and upper bounds on minimax risks agree up to constant factors when p=2, which together provide the information-theoretic limits of estimating a sparse vector in the high-dimensional linear errors-in-variables model. An estimator for the underlying parameter is also proposed and shown to be minimax optimal in the ℓ2-loss.

Rate of convergence to equilibrium of marked Hawkes processes

2002 ◽  
Vol 39 (1) ◽  
pp. 123-136 ◽  
Author(s):  
P. Brémaud ◽  
G. Nappo ◽  
G. L. Torrisi
Keyword(s):  
Rate Of Convergence ◽  
Stationary Process ◽  
Rates Of Convergence ◽  
Stopping Rules ◽  
Convergence To Equilibrium ◽  
Situation Models ◽  
Hawkes Process ◽  
Hawkes Processes ◽  
Simulation Algorithms

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

ON THE SPEED OF CONVERGENCE OF MULTIDIMENSIONAL DIFFUSIONS TO EQUILIBRIUM

2012 ◽  
Vol 12 (01) ◽  
pp. 1150003 ◽  
Author(s):  
RABI BHATTACHARYA ◽  
ARAMIAN WASIELAK
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Rates Of Convergence ◽  
Speed Of Convergence ◽  
Discrete Parameter ◽  
Return Times ◽  
Coupling Techniques ◽  
Multidimensional Diffusions ◽  
New Criteria

We obtain new criteria for polynomial rates of convergence of ergodic multidimensional diffusions to equilibrium. For this, (1) a method is provided for adapting to continuous-time Markov processes coupling techniques for discrete parameter Harris processes, and (2) estimates of moments of return times to a ball are derived.

scholarly journals Rates of convergence in the two-island and isolation-with-migration models

2021 ◽  
Author(s):  
Brandon Legried ◽  
Jonathan Terhorst
Keyword(s):  
Lower Bounds ◽  
Rates Of Convergence ◽  
Upper Bounds ◽  
Concentration Of Measure ◽  
Upper And Lower Bounds ◽  
Information Theoretic ◽  
Isolation With Migration ◽  
Demographic Inference ◽  
And Migration ◽  
Inference Methods

AbstractA number of powerful demographic inference methods have been developed in recent years, with the goal of fitting rich evolutionary models to genetic data obtained from many populations. In this paper we investigate the statistical performance of these methods in the specific case where there is continuous migration between populations. Compared with earlier work, migration significantly complicates the theoretical analysis and demands new techniques. We employ the theories of phase-type distributions and concentration of measure in order to study the two-island and isolation-with-migration models, resulting in both upper and lower bounds. For the upper bounds, we consider inferring rates of coalescent and migration on the basis of directly observing pairwise coalescent times, and, more realistically, when (conditionally) Poisson-distributed mutations dropped on latent trees are observed. We complement these upper bounds with information-theoretic lower bounds which establish a limit, in terms of sample size, below which inference is effectively impossible.

Rate of convergence to equilibrium of marked Hawkes processes

2002 ◽  
Vol 39 (01) ◽  
pp. 123-136 ◽  
Author(s):  
P. Brémaud ◽  
G. Nappo ◽  
G. L. Torrisi
Keyword(s):  
Rate Of Convergence ◽  
Stationary Process ◽  
Rates Of Convergence ◽  
Stopping Rules ◽  
Convergence To Equilibrium ◽  
Situation Models ◽  
Hawkes Process ◽  
Hawkes Processes ◽  
Simulation Algorithms

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

scholarly journals Subgeometric rates of convergence for Markov processes under subordination

2017 ◽  
Vol 49 (1) ◽  
pp. 162-181 ◽  
Author(s):  
Chang-Song Deng ◽  
René L. Schilling ◽  
Yan-Hong Song
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Convergence Rate ◽  
Rate Of Convergence ◽  
Rates Of Convergence ◽  
Bernstein Function ◽  
Convergence To Equilibrium ◽  
Speed Of Convergence ◽  
Crucial Point ◽  
Original Process

Abstract We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

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