MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
I do not know if this helps:… – András Bátkai Jul 19 '13 at 16:46
up vote 9 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.