# On Stability and Trajectory Boundedness in Mean-square Sense for ARMA Processes

On Stability and Trajectory Boundedness in Mean-square Sense for ARMA Processes For the multidimensional ARMA system A(z)y k = C(z)w k it is shown that stability (det A(z) ≠= 0, ∀ z : |z| ≤ 1) of A(z) is equivalent to the trajectory boundedness in the mean square sense (MSS) $${\mathop {\lim {\kern 1pt} {\kern 1pt} \sup }\limits_{n \to \infty } }{\kern 1pt} \frac{1} {n}{\sum\limits_{k = 1}^n {{\left\| {y_{k} } \right\|}^{2} < \infty \;\;{\kern 1pt} {\kern 1pt} {\text{a}}{\text{.s}}.,{\kern 1pt} } }$$ which, as a rule, is a consequence of a successful stochastic adaptive control leading the closed-loop of an ARMAX system to a steady state ARMA system. In comparison with existing results the stability condition imposed on C(z) is no longer needed. The only structural requirement on the system is that det A(z) and det C(z) have no unstable common factor. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# On Stability and Trajectory Boundedness in Mean-square Sense for ARMA Processes

, Volume 19 (4) – Nov 2, 2015
8 pages

/lp/springer-journals/on-stability-and-trajectory-boundedness-in-mean-square-sense-for-arma-gp3j06PT6T
Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-003-0132-0
Publisher site
See Article on Publisher Site

### Abstract

For the multidimensional ARMA system A(z)y k = C(z)w k it is shown that stability (det A(z) ≠= 0, ∀ z : |z| ≤ 1) of A(z) is equivalent to the trajectory boundedness in the mean square sense (MSS) $${\mathop {\lim {\kern 1pt} {\kern 1pt} \sup }\limits_{n \to \infty } }{\kern 1pt} \frac{1} {n}{\sum\limits_{k = 1}^n {{\left\| {y_{k} } \right\|}^{2} < \infty \;\;{\kern 1pt} {\kern 1pt} {\text{a}}{\text{.s}}.,{\kern 1pt} } }$$ which, as a rule, is a consequence of a successful stochastic adaptive control leading the closed-loop of an ARMAX system to a steady state ARMA system. In comparison with existing results the stability condition imposed on C(z) is no longer needed. The only structural requirement on the system is that det A(z) and det C(z) have no unstable common factor.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Nov 2, 2015

### References

Access the full text.