Point process convergence of stochastic volatility processes with application to sample autocorrelation

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

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Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

Advances in Applied Probability ◽

10.1239/aap/1354716591 ◽

2012 ◽

Vol 44 (4) ◽

pp. 1113-1141 ◽

Cited By ~ 5

Author(s):

Rafał Kulik ◽

Philippe Soulier

Keyword(s):

Asymptotic Behavior ◽

Stochastic Volatility ◽

Long Memory ◽

Convergence Rates ◽

Partial Sums ◽

Infinite Variance ◽

Tail Behavior ◽

Stochastic Volatility Models ◽

Crucial Difference ◽

Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

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Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

Advances in Applied Probability ◽

10.1017/s0001867800006066 ◽

2012 ◽

Vol 44 (04) ◽

pp. 1113-1141

Author(s):

Rafał Kulik ◽

Philippe Soulier

Keyword(s):

Asymptotic Behavior ◽

Stochastic Volatility ◽

Long Memory ◽

Convergence Rates ◽

Partial Sums ◽

Infinite Variance ◽

Tail Behavior ◽

Stochastic Volatility Models ◽

Crucial Difference ◽

Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

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ASYMPTOTIC PROPERTIES OF KOLMOGOROV WIDTHS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972710000043 ◽

2010 ◽

Vol 82 (1) ◽

pp. 10-17

Author(s):

MIKHAIL I. OSTROVSKII

Keyword(s):

Banach Space ◽

Asymptotic Behavior ◽

Banach Spaces ◽

Asymptotic Properties ◽

Codimension One ◽

Kolmogorov Widths

AbstractWe consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

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Asymptotic Behavior of a Viscoelastic Fluid in a Closed Loop Thermosyphon: Physical Derivation, Asymptotic Analysis, and Numerical Experiments

Abstract and Applied Analysis ◽

10.1155/2013/748683 ◽

2013 ◽

Vol 2013 ◽

pp. 1-20 ◽

Cited By ~ 4

Author(s):

Justine Yasappan ◽

Ángela Jiménez-Casas ◽

Mario Castro

Keyword(s):

Asymptotic Behavior ◽

Viscoelastic Fluid ◽

Closed Loop ◽

Theoretical Study ◽

Equations Of Motion ◽

Asymptotic Properties ◽

Inertial Manifold ◽

Thermal Gradients ◽

External Temperature ◽

First Time

Fluids subject to thermal gradients produce complex behaviors that arise from the competition with gravitational effects. Although such sort of systems have been widely studied in the literature for simple (Newtonian) fluids, the behavior of viscoelastic fluids has not been explored thus far. We present a theoretical study of the dynamics of a Maxwell viscoelastic fluid in a closed-loop thermosyphon. This sort of fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and the exact equations of motion in the inertial manifold that characterizes the asymptotic behavior. We derive, for the first time, the mathematical derivations of the motion of a viscoelastic fluid in the interior of a closed-loop thermosyphon under the effects of natural convection and a given external temperature gradient.

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Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

Advances in Applied Probability ◽

10.1017/s0001867800007229 ◽

2014 ◽

Vol 46 (02) ◽

pp. 585-602 ◽

Cited By ~ 1

Author(s):

Li-Xin Zhang ◽

Feifang Hu ◽

Siu Hung Cheung ◽

Wai Sum Chan

Keyword(s):

Asymptotic Behavior ◽

Rate Of Convergence ◽

Clinical Studies ◽

Asymptotic Properties ◽

Important Application ◽

Urn Models ◽

Theoretical Justification ◽

Treatment Allocation ◽

Number Of Patients ◽

Generalized Pólya Urn

The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.

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Functional Linear Regression

10.1093/oxfordhb/9780199568444.013.2 ◽

2018 ◽

Author(s):

Hervé Cardot ◽

Pascal Sarda

Keyword(s):

Linear Regression ◽

Least Squares ◽

Linear Models ◽

Estimation Error ◽

Asymptotic Properties ◽

Principal Component Regression ◽

Principal Component ◽

Penalized Least Squares ◽

Open Problems ◽

Functional Linear Regression

This article presents a selected bibliography on functional linear regression (FLR) and highlights the key contributions from both applied and theoretical points of view. It first defines FLR in the case of a scalar response and shows how its modelization can also be extended to the case of a functional response. It then considers two kinds of estimation procedures for this slope parameter: projection-based estimators in which regularization is performed through dimension reduction, such as functional principal component regression, and penalized least squares estimators that take into account a penalized least squares minimization problem. The article proceeds by discussing the main asymptotic properties separating results on mean square prediction error and results on L2 estimation error. It also describes some related models, including generalized functional linear models and FLR on quantiles, and concludes with a complementary bibliography and some open problems.

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Limit Theory for High Frequency Sampled MCARMA Models

Advances in Applied Probability ◽

10.1239/aap/1409319563 ◽

2014 ◽

Vol 46 (3) ◽

pp. 846-877 ◽

Cited By ~ 3

Author(s):

Vicky Fasen

Keyword(s):

Asymptotic Behavior ◽

High Frequency ◽

Lévy Process ◽

Asymptotic Properties ◽

Domain Of Attraction ◽

Levy Process ◽

Sample Variance ◽

Limit Theory ◽

Second Moments ◽

Time Grid

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

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Generalized Linear Models for Point Process Analyses of Neural Spiking Activity

Encyclopedia of Computational Neuroscience ◽

10.1007/978-1-4614-6675-8_393 ◽

2015 ◽

pp. 1284-1287

Author(s):

Zhe Chen ◽

Emery N. Brown

Keyword(s):

Point Process ◽

Generalized Linear Models ◽

Linear Models ◽

Neural Spiking ◽

Spiking Activity

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Robust Statistical Inference in Generalized Linear Models Based on Minimum Renyi’s Pseudodistance Estimators

Entropy ◽

10.3390/e24010123 ◽

2022 ◽

Vol 24 (1) ◽

pp. 123

Author(s):

María Jaenada ◽

Leandro Pardo

Keyword(s):

Generalized Linear Models ◽

Influence Function ◽

Regression Models ◽

Linear Models ◽

Asymptotic Properties ◽

Significant Loss ◽

Superior Performance ◽

Test Statistics ◽

Linear Regression Models ◽

Type Test

Minimum Renyi’s pseudodistance estimators (MRPEs) enjoy good robustness properties without a significant loss of efficiency in general statistical models, and, in particular, for linear regression models (LRMs). In this line, Castilla et al. considered robust Wald-type test statistics in LRMs based on these MRPEs. In this paper, we extend the theory of MRPEs to Generalized Linear Models (GLMs) using independent and nonidentically distributed observations (INIDO). We derive asymptotic properties of the proposed estimators and analyze their influence function to asses their robustness properties. Additionally, we define robust Wald-type test statistics for testing linear hypothesis and theoretically study their asymptotic distribution, as well as their influence function. The performance of the proposed MRPEs and Wald-type test statistics are empirically examined for the Poisson Regression models through a simulation study, focusing on their robustness properties. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the superior performance of the robust MRPEs as well as Wald-type tests.

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