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!!



Anyone has Kushner's book "Introduction to stochastic control" 1971? I need a theorem from it
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.


