scholarly journals Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

2019 ◽  
Vol 23 ◽  
pp. 803-822
Author(s):  
Mikkel Slot Nielsen ◽  
Jan Pedersen
Keyword(s):  
Quadratic Form ◽  
Continuous Time ◽  
Quadratic Forms ◽  
Statistical Estimation ◽  
Limit Theorems ◽  
Stationary Process ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Stationary Processes ◽  
Limiting Behavior

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

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.

scholarly journals On a Necessary Condition for the Sample Path Continuity of Weakly Stationary Processes

1970 ◽  
Vol 38 ◽  
pp. 103-111 ◽  
Author(s):  
Izumi Kubo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Spectral Distribution ◽  
Covariance Function ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Sample Path ◽  
Necessary Condition ◽  
Stationary Processes ◽  
Spectral Distribution Function

We shall discuss the sample path continuity of a stationary process assuming that the spectral distribution function F(λ) is given. Many kinds of sufficient conditions have been given in terms of the covariance function or the asymptotic behavior of the spectral distribution function.

Conditions for strong ergodicity using intensity matrices

1988 ◽  
Vol 25 (1) ◽  
pp. 34-42 ◽  
Author(s):  
Jean Johnson ◽  
Dean Isaacson
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Limiting Behavior ◽  
Strong Ergodicity ◽  
Continuous Time Markov Chains

Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψn of Pn(ψ nPn = ψ n) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.

Noncausality in Continuous Time Models

1996 ◽  
Vol 12 (2) ◽  
pp. 215-256 ◽  
Author(s):  
F. Comte ◽  
E. Renault
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Stochastic Volatility Models ◽  
Discrete Sample ◽  
Volatility Models ◽  
Moving Average Representation ◽  
Canonical Representations ◽  
Average Representation

In this paper, we study new definitions of noncausality, set in a continuous time framework, illustrated by the intuitive example of stochastic volatility models. Then, we define CIMA processes (i.e., processes admitting a continuous time invertible moving average representation), for which canonical representations and sufficient conditions of invertibility are given. We can provide for those CIMA processes parametric characterizations of noncausality relations as well as properties of interest for structural interpretations. In particular, we examine the example of processes solutions of stochastic differential equations, for which we study the links between continuous and discrete time definitions, find conditions to solve the possible problem of aliasing, and set the question of testing continuous time noncausality on a discrete sample of observations. Finally, we illustrate a possible generalization of definitions and characterizations that can be applied to continuous time fractional ARMA processes.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (03) ◽  
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 {X n , n ≥ 0} and two observables, τ(∙) and V(∙), 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 {X n , n ≥ 0} is a sequence of independent and identically distributed random variables.

Quadratic forms positive definite on a linear manifold

1952 ◽  
Vol 48 (1) ◽  
pp. 70-71
Author(s):  
L. S. Goddard
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. In a recent paper(1), Afriat has given necessary and sufficient conditions for a real quadratic form to be positive definite on a linear manifold, in terms of the dual Grassmannian coordinates of the manifold. Considerable matrix manipulations were used in Afriat's method, but most of these may be avoided by the method of the present paper, which depends on some well-known properties of the Grassmannian coordinates. We first show that the conditions may be expressed as a set of inequalities which are quadratic in the Grassmannian coordinates of the manifold. Then, by a standard theorem, these may be transformed into Afriat's conditions on the dual coordinates.

High-Level exceedances of regenerative and semi-stationary processes

1980 ◽  
Vol 17 (02) ◽  
pp. 423-431 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Point Processes ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Partial Sums ◽  
Stationary Processes ◽  
Special Cases ◽  
Cumulative Amount ◽  
High Level ◽  
Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

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