Rate of convergence to equilibrium of marked Hawkes processes

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.

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Subgeometric rates of convergence for Markov processes under subordination

Advances in Applied Probability ◽

10.1017/apr.2016.83 ◽

2017 ◽

Vol 49 (1) ◽

pp. 162-181 ◽

Cited By ~ 2

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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UNIFORM ASYMPTOTIC NORMALITY IN STATIONARY AND UNIT ROOT AUTOREGRESSION

Econometric Theory ◽

10.1017/s0266466611000016 ◽

2011 ◽

Vol 27 (6) ◽

pp. 1117-1151 ◽

Cited By ~ 16

Author(s):

Chirok Han ◽

Peter C. B. Phillips ◽

Donggyu Sul

Keyword(s):

Normal Distribution ◽

Rate Of Convergence ◽

Unit Root ◽

Asymptotic Theory ◽

Signal Strength ◽

Rates Of Convergence ◽

Moment Conditions ◽

First Order ◽

The Cost ◽

Artificial Shrinkage

While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution that holds uniformly in the autoregressive coefficient ρ, including stationary and unit root cases. The rate of convergence is $\root \of n $ when $\left| \rho \right| < 1$ and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when ρ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator (FAE) is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases, which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity. Confidence intervals constructed from the FAE using local asymptotic theory around unity also lead to improvements over the MLE.

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Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise

Annales de l Institut Henri Poincaré Probabilités et Statistiques ◽

10.1214/15-aihp724 ◽

2017 ◽

Vol 53 (2) ◽

pp. 503-538

Author(s):

Joaquin Fontbona ◽

Fabien Panloup

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Rate Of Convergence ◽

Multiplicative Noise ◽

Convergence To Equilibrium

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Acceptability ratings cannot be taken at face value

Linguistic Intuitions ◽

10.1093/oso/9780198840558.003.0011 ◽

2020 ◽

pp. 189-214

Author(s):

Carson T. Schütze

Keyword(s):

Rate Of Convergence ◽

Experimental Work ◽

Rates Of Convergence ◽

Original Method ◽

Journal Articles ◽

True Rate ◽

Acceptability Judgments ◽

Critical Words ◽

Recent Experimental Work

This chapter addresses how linguists’ empirical (syntactic) claims should be tested with non-linguists. Recent experimental work attempts to measure rates of convergence between data presented in journal articles and the results of large surveys. Three follow-up experiments to one such study are presented. It is argued that the original method may underestimate the true rate of convergence because it leaves considerable room for naïve subjects to give ratings that do not reflect their true acceptability judgments of the relevant structures. To understand what can go wrong, the experiments were conducted in two parts. The first part had visually presented sentences rated on a computer, replicating previous work. The second part was an interview where the experimenter asked the participants about the ratings they gave to particular items, in order to determine what interpretation or parse they had assigned, whether they had missed any critical words, and so on.

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The rate of convergence of extremes of stationary normal sequences

Advances in Applied Probability ◽

10.1017/s0001867800020991 ◽

1983 ◽

Vol 15 (01) ◽

pp. 54-80 ◽

Cited By ~ 2

Author(s):

Holger Rootzén

Keyword(s):

Rate Of Convergence ◽

Joint Distribution ◽

Point Processes ◽

Rates Of Convergence ◽

Joint Distribution Function ◽

Maximal Correlation ◽

Standard Normal ◽

Image Position ◽

The Right ◽

Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

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A class of interacting marked point processes: rate of convergence to equilibrium

Journal of Applied Probability ◽

10.1239/jap/1019737994 ◽

2002 ◽

Vol 39 (1) ◽

pp. 137-160 ◽

Cited By ~ 2

Author(s):

G. L. Torrisi

Keyword(s):

Stochastic Processes ◽

Rate Of Convergence ◽

Point Processes ◽

Marked Point ◽

Marked Point Processes ◽

Convergence To Equilibrium ◽

Birth And Death Processes ◽

Loss Networks

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

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Asymmetries in financial returns

International Journal of Financial Engineering ◽

10.1142/s2424786317500451 ◽

2017 ◽

Vol 04 (04) ◽

pp. 1750045 ◽

Cited By ~ 13

Author(s):

Dilip B. Madan ◽

King Wang

Keyword(s):

Time Series ◽

Rate Of Convergence ◽

Time Series Data ◽

Gamma Process ◽

Series Data ◽

Convergence To Equilibrium ◽

Financial Returns ◽

Delta Hedging ◽

The Difference

Market clichés assert that markets take escalators up and elevators down. The observation suggests differentiating models for up and down moves. Non-diffusive models allow for this and we model the move as the difference of two independent mean reverting increasing processes driven by gamma process shocks. The model is estimated on time series data as well as option data. Broadly speaking, the rise occurs with more frequent and smaller jumps with a faster rate of convergence to equilibrium. The down tick process has larger, less frequent moves with longer memories. Applications to delta hedging and the setting of profit targets and stop losses are also presented.

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