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.

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.

The condition for extinction by probability one in a birth, death and migration process

1975 ◽  
Vol 7 (01) ◽  
pp. 61-65
Author(s):  
Inge S. Helland
Keyword(s):  
Extinction Probability ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Migration Process ◽  
And Migration ◽  
Necessary And Sufficient ◽  
Birth Death

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

The condition for extinction by probability one in a birth, death and migration process

1975 ◽  
Vol 7 (1) ◽  
pp. 61-65 ◽  
Author(s):  
Inge S. Helland
Keyword(s):  
Extinction Probability ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Migration Process ◽  
And Migration ◽  
Necessary And Sufficient ◽  
Birth Death

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

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.

Stochastic monotonicity of birth–death processes

1980 ◽  
Vol 12 (01) ◽  
pp. 59-80 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Initial Distribution ◽  
Unit Vector ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Initial State ◽  
Distribution Vector ◽  
Necessary And Sufficient ◽  
Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t 1, t 2) if Pr {x(t) > i} is increasing (decreasing) with t on (t 1, t 2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

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 Weak convergence of conditioned birth-death processes in discrete time

1997 ◽  
Vol 34 (01) ◽  
pp. 46-53
Author(s):  
Pauline Schrijner ◽  
Erik A. Van Doorn
Keyword(s):  
Weak Convergence ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Limiting Behaviour ◽  
Necessary And Sufficient ◽  
Birth Death

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

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.

Stochastic monotonicity of birth–death processes

1980 ◽  
Vol 12 (1) ◽  
pp. 59-80 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Initial Distribution ◽  
Unit Vector ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Initial State ◽  
Distribution Vector ◽  
Necessary And Sufficient ◽  
Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

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