Weak Convergence to Stochastic Integrals

2021 ◽  
pp. 724-756
Author(s):  
James Davidson
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Final Section ◽  
Stochastic Integrals ◽  
Martingale Difference ◽  
Mixing Processes ◽  
Linear Processes ◽  
Special Cases

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS I

2000 ◽  
Vol 16 (5) ◽  
pp. 621-642 ◽  
Author(s):  
Robert M. de Jong ◽  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Unit Root ◽  
Functional Central Limit Theorem ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Unit Root Testing ◽  
Functional Central Limit

This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS II

2000 ◽  
Vol 16 (5) ◽  
pp. 643-666 ◽  
Author(s):  
James Davidson ◽  
Robert M. de Jong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Functional Central Limit Theorem ◽  
Partial Sums ◽  
Stochastic Integrals ◽  
Mixing Processes ◽  
Convergence Results ◽  
Functional Central Limit

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

Bivariate Pseudo-Spectral Local Linearisation Method for Mixed Convective Flow Over the Vertical Frustum of a Cone in a Nanofluid with Soret and Viscous Dissipation Effects

2017 ◽  
Vol 33 (5) ◽  
pp. 687-702
Author(s):  
Ch. RamReddy ◽  
Ch. Venkata Rao
Keyword(s):  
Mass Transfer ◽  
Brownian Motion ◽  
Spectral Method ◽  
Viscous Dissipation ◽  
Surface Drag ◽  
Special Cases ◽  
Soret Number ◽  
Uniform Wall ◽  
Mixed Convective Flow ◽  
Pseudo Spectral

AbstractIn this investigation, we intend to present the influence of the prominent viscous dissipation and Soret effects on mixed convection heat and mass transfer over the vertical frustum of a cone in a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition, the uniform wall nanoparticle condition at the surface is replaced with the zero nanoparticle mass flux condition to execute physically applicable results. The governing equations of a nanofluid flow in the dimensional form are reduced to a system of partial differential equations in the non-dimensional form by using suitable non-similarity variables and then solved by using a recently introduced spectral method named as Bivariate Pseudo-Spectral Local Linearisation Method (BPSLLM). The convergence and error analysis tests are conducted to examine the accuracy of the spectral method. To validate the method, the present numerical results are compared with the existing results in some special cases and the outcomes are observed to be in very good agreement. The effects of Brownian motion, thermophoresis, Eckert number, Soret number, nanoparticle and regular buoyancy parameters on the dimensionless surface drag, heat, nanoparticle mass and regular mass transfer rates over the vertical frustum of a cone are discussed and illustrated graphically.

scholarly journals Learning Theory Estimates with Observations from General Stationary Stochastic Processes

2016 ◽  
Vol 28 (12) ◽  
pp. 2853-2889 ◽  
Author(s):  
Hanyuan Hang ◽  
Yunlong Feng ◽  
Ingo Steinwart ◽  
Johan A. K. Suykens
Keyword(s):  
Stochastic Processes ◽  
Type Inequality ◽  
Empirical Risk Minimization ◽  
Risk Minimization ◽  
Mixing Processes ◽  
Gaussian Kernels ◽  
Empirical Risk ◽  
Learning Rates ◽  
Stationary Stochastic Processes ◽  
Learning Schemes

This letter investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by general, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a unified treatment on analyzing the learning schemes with various mixing processes can be conducted and a sharp oracle inequality for generic regularized empirical risk minimization schemes can be established. The obtained oracle inequality is then applied to derive convergence rates for several learning schemes such as empirical risk minimization (ERM), least squares support vector machines (LS-SVMs) using given generic kernels, and SVMs using gaussian kernels for both least squares and quantile regression. It turns out that for independent and identically distributed (i.i.d.) processes, our learning rates for ERM recover the optimal rates. For non-i.i.d. processes, including geometrically [Formula: see text]-mixing Markov processes, geometrically [Formula: see text]-mixing processes with restricted decay, [Formula: see text]-mixing processes, and (time-reversed) geometrically [Formula: see text]-mixing processes, our learning rates for SVMs with gaussian kernels match, up to some arbitrarily small extra term in the exponent, the optimal rates. For the remaining cases, our rates are at least close to the optimal rates. As a by-product, the assumed generalized Bernstein-type inequality also provides an interpretation of the so-called effective number of observations for various mixing processes.

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