Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 1 ,pp. 12--35

STABILITY AND FUNCTIONAL LIMIT THEOREMS FOR RANDOM DEGENERATE DIFFUSIONS

By

GOPAL K. BASAK , ARNAB BISI
* Hong Kong University of Science and Technology, Kowloon*

and

MRINAL K. GHOSH
* Indian Institute of Science, Bangalore*

*SUMMARY.*We study the stability and functional limit theorems for a class
of random degenerate diffusions
where the flow is driven by a Wiener process and an independent
Markov chain. Under a Liapunov type condition
we establish certain growth properties and asymptotic flatness of
the flow. This yields the existence of a unique invariant probability
p
and stability in distribution. We then identify a broad subset of
L^{2}($\RR$^{d} * Q,p) which
belongs to the range of the infinitesimal generator of the random
diffusion.
For functions in this set we derive the functional central limit
theorem and
the law of iterated logarithm

*AMS (1991) subject classification.* 60F17, 60J60.

*Key words and phrases. *Random diffusions, Markov chain, tightness,
stability, functional central limit theorem, law of iterated logarithm.