FOURTH MOMENT STRUCTURE OF THE GARCH(p,q) PROCESS

1999 ◽  
Vol 15 (6) ◽  
pp. 824-846 ◽  
Author(s):  
Changli He ◽  
Timo Teräsvirta
Keyword(s):  
Statistical Theory ◽  
Autocorrelation Function ◽  
Fourth Moment ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
The Moment ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Equivalent representations of max-stable processes via ℓp-norms

2018 ◽  
Vol 55 (1) ◽  
pp. 54-68
Author(s):  
Marco Oesting
Keyword(s):  
Spectral Representation ◽  
Stable Process ◽  
Tail Dependence ◽  
Stable Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dependence Function ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Stable Tail Dependence Function

Abstract While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on ℓp-norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of an ℓp-norm-based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.

scholarly journals C*-Algebras of Irreversible Dynamical Systems

2006 ◽  
Vol 58 (1) ◽  
pp. 39-63 ◽  
Author(s):  
R. Exel ◽  
A. Vershik
Keyword(s):  
Measure Space ◽  
General Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Circle Actions ◽  
Generators And Relations ◽  
C Algebras ◽  
Special Cases ◽  
Measure Preserving Transformation ◽  
Necessary And Sufficient

AbstractWe show that certain C*-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

A birth, death and migration process by immigration

1975 ◽  
Vol 7 (01) ◽  
pp. 44-60
Author(s):  
M. Aksland
Keyword(s):  
Sufficient Condition ◽  
Algebraic Equations ◽  
Necessary And Sufficient Condition ◽  
Migration Process ◽  
Immigration Process ◽  
Special Cases ◽  
And Migration ◽  
Necessary And Sufficient ◽  
Sequence Of Functions ◽  
Birth Death

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies. Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.

scholarly journals Biwarped product submanifolds of a Kähler manifold

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2349-2365 ◽  
Author(s):  
Hakan Taştan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Kähler Manifold ◽  
Second Fundamental Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Necessary And Sufficient

We study biwarped product submanifolds which are special cases of multiply warped product submanifolds in K?hler manifolds. We observe the non-existence of such submanifolds under some circumstances. We show that there exists a non-trivial biwarped product submanifold of a certain type by giving an illustrate example. We also give a necessary and sufficient condition for such submanifolds to be locally trivial. Moreover, we establish an inequality for the squared norm of the second fundamental form in terms of the warping functions for such submanifolds. The equality case is also discussed.

A birth, death and migration process by immigration

1975 ◽  
Vol 7 (1) ◽  
pp. 44-60 ◽  
Author(s):  
M. Aksland
Keyword(s):  
Sufficient Condition ◽  
Algebraic Equations ◽  
Necessary And Sufficient Condition ◽  
Migration Process ◽  
Immigration Process ◽  
Special Cases ◽  
And Migration ◽  
Necessary And Sufficient ◽  
Sequence Of Functions ◽  
Birth Death

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies.Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.

scholarly journals The moment spaces of normed linear spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 277-283
Author(s):  
Fowzi Ahmad Sejeeni ◽  
Matooq Ahmad Badri
Keyword(s):  
Normed Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Independent Sequence ◽  
Linear Spaces ◽  
Linearly Independent ◽  
The Moment ◽  
Moment Spaces ◽  
Necessary And Sufficient

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

scholarly journals Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

2021 ◽  
Author(s):  
Hristo Manev ◽  
Mancho Manev
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Special Cases ◽  
Necessary And Sufficient

It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.

On the general bilinear time series model

1988 ◽  
Vol 25 (03) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Separability Conditions for a Subclass of Circulant States in Two-Qutrit System and Quasi-Distillation of Entanglement

2013 ◽  
Vol 20 (03) ◽  
pp. 1340006
Author(s):  
Tatsunori Saito ◽  
Masanari Asano ◽  
Takashi Matsuoka ◽  
Masanori Ohya
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Failure Condition ◽  
Special Cases ◽  
Separability Condition ◽  
Necessary And Sufficient

We will show the necessary and sufficient condition of separability for a sub-class of circulant states in the ℂ3 ⊗ ℂ3 system. This sub-class includes the models proposed by several authors as special cases. Moreover, it will be proved that our separability condition equals to the failure condition of quasi-distillation of entanglement.

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