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.