In a paper I'm reading, it refers to Theorem 8, Page 217 of the book
"Introduction to Stochastic Control" H. J. Kushner, New York: Holt, Reinhart, and Winston 1971. Unfortunately I don't have it and the copy in our library was checked out.
Does anyone here happen to have that book at hand and let me know what the theorem says?
It's a stochastic version of LaSalle's Theorem.
Thanks!!
The Carleton College library has a copy of the Kushner book. Here's the theorem:
Theorem 8
Let* $P\gt0, C\ge0$ and
$$EA_n'PA_n-P=-C.\ \ \ (8.24)$$
Then $EX_n'CX_n\rightarrow0$ and $X_n'CX_n\rightarrow0$ w.p.l. Also
$$P_x(\sup_{\infty\gt n\ge0} X_n'PX_n \ge \lambda) \le {x'Px\over\lambda}.$$
Hence, the measures $\mu_n$ (corresponding to $X_n$) are weakly bounded. Then $X_n$ converges (in probability) to the support of the largest invariant set whose support is contained in $L=\{x: x'Cx=0\}$.
Let the $A_n$ be identically distributed. If $\{ X_n \}$ is mean square stable (that is, $EX_n'X_n\rightarrow0$), then for any $C\gt0$, there is a $P\gt0$ satisfying $(8.24)$. Next, consider the operation $(8.24)$ as a linear equation in the components $p_{ij}$ of $P$, and let $Q$ and $D$ denote the vectors composed of ordering the matrices $P$ and $C$ into vectors. Then write $(8.24)$ as $BQ=-D$. A necessary and sufficient condition for mean square stability (and sufficient for w.p.l. stability) is that the eigenvalues of $B$ lie in the unit circle.
*: $A\gt0$ means positive definite. $A\ge0$ means positive semidefinite.
[[my note: In the original, the vectors and matrix I've called $Q$, $D$, and $B$ appear as script versions of $P$, $C$, and $A$. I couldn't easily figure out the TeX equivalent, so I did the next best thing.]]
I hope this helps.
