Existence of moments in a stationary stochastic difference equation

1990 ◽  
Vol 22 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Hans Arnfinn Karlsen
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Difference Equation ◽  
Dependent Variables ◽  
Gaussian Case ◽  
Necessary And Sufficient

The stationary stochastic difference equation Xt = YtXt–1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt||p <∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt)}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

Existence of moments in a stationary stochastic difference equation

1990 ◽  
Vol 22 (01) ◽  
pp. 129-146 ◽  
Author(s):  
Hans Arnfinn Karlsen
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Difference Equation ◽  
Dependent Variables ◽  
Gaussian Case ◽  
Necessary And Sufficient

The stationary stochastic difference equation Xt = YtXt –1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt || p &lt;∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt )}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

scholarly journals On the attraction to the normal law of functions of weakly dependent variables

2020 ◽  
pp. 24-39
Author(s):  
A. G. Grin
Keyword(s):  
Symmetric Functions ◽  
Sufficient Conditions ◽  
Weak Dependence ◽  
Necessary And Sufficient Conditions ◽  
Dependent Variables ◽  
Normal Law ◽  
Necessary And Sufficient ◽  
Weakly Dependent

The paper gives a “generally accepted” condition of weak dependence and a condition on the class of symmetric functions that provide fulfilment of the necessary and sufficient conditions, obtained earlier by the author, for attracting functions of dependent quantities to the normal law.

scholarly journals Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

2015 ◽  
Vol 3 (2) ◽  
pp. 61
Author(s):  
A. Murgesan ◽  
P. Sowmiya
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference

<p>In this paper, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the first order neutral delay difference equation with constant coefficients of the form <br />\begin{equation*} \quad \quad \quad \quad \Delta[x(n)-px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0 \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}<br />by constructing several suitable auxiliary functions. Some examples are also given to illustrate our results.</p>

scholarly journals Asymptotic Behavior of Solutions to Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

scholarly journals Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Guoqiang Dang ◽  
Jinhua Cai
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
System Of Equations ◽  
Entire Solutions ◽  
Differential Difference Equation ◽  
Necessary And Sufficient

In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation f″z2+△ckfz2=1 and the system of equations f1″z2+△ckf2z2=1 and f2″z2+△ckf1z2=1 have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

scholarly journals Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2751-2770
Author(s):  
Aleksandra Kapesic ◽  
Jelena Manojlovic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Order Difference ◽  
Varying Coefficients ◽  
Regularly Varying ◽  
Necessary And Sufficient

Positive decreasing solutions of the nonlinear difference equation ?(pn|?xn|?-1?xn)=qn|xn+1|?-1xn+1, n ? 1, ? > ? > 0, are studied under the assumption that p; q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

A Liapunov functional for a scalar differential difference equation

1978 ◽  
Vol 79 (3-4) ◽  
pp. 307-316 ◽  
Author(s):  
E. F. Infante ◽  
J. A. Walker
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quadratic Functional ◽  
Liapunov Functional ◽  
Differential Difference Equation ◽  
The Difference ◽  
Rates Of Decay ◽  
Necessary And Sufficient

SynopsisGiven the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

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