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.

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.

MULTIVARIATE COMPOSITE COPULAS

2021 ◽  
pp. 1-40
Author(s):  
Jiehua Xie ◽  
Jun Fang ◽  
Jingping Yang ◽  
Lan Bu
Keyword(s):  
Tail Dependence ◽  
Financial Data ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Composite Function ◽  
Probability Structure ◽  
Necessary And Sufficient ◽  
Bivariate Functions ◽  
New Framework ◽  
Multivariate Copula

Abstract In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

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 On Approximation of the Perturbed Inclusion

2007 ◽  
Vol 14 (2) ◽  
pp. 253-267
Author(s):  
Alexander I. Bulgakov ◽  
Anna A. Grigorenko ◽  
Anatoliy I. Korobko
Keyword(s):  
Stable Process ◽  
Approximate Solutions ◽  
Continuous Functions ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Linear Integral ◽  
The Right ◽  
Necessary And Sufficient

Abstract The paper is concerned with the so-called perturbed inclusion in the space of continuous functions. The right-hand side of the inclusion is represented by an algebraic sum of the values of two multi-valued maps, one of which consists of compacts and the other is not necessarily closed-valued and is a composition of a linear integral operator and multimap convex-valued with respect to switching. For such an inclusion it is proved that approximation in the space of summable functions of the values of a multimap convex-valued with respect to switching is not always a stable process. The necessary and sufficient condition for the closure of the set of approximate solutions to converge to the closure of the set of solutions for perturbed inclusion is derived.

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 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.

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