Central limit theorems for random evolutions
In a series of papers [ibid. 19, 183-187 (1985; Zbl 0568.60007), ibid. 43, No. 2, 363-365 (1992), Stat. Probab. Lett. 6, No. 4, 225-227 (1988; Zbl 0635.60082), Acta Math. Hung. 57, No. 1/2, 3-5 (1991; Zbl 0757.60064) and Stochastic Processes Appl. 37, No. 1, 99-116 (1991; Zbl 0724.60010)] the author used a product formula of \textit{P. R. Chernoff} [J. Funct. Anal. 2, 238-242 (1968; Zbl 0157.215)] for linear operators to get central limit theorems for sums of dependent random variables and for Markov chains. Here the author derives an exponential formula of the form \[ \bigl( I + A/n + B/n^ 2 + C/n^ 3 \bigr)^{n^ 3}f = \bigl( e^{nG_ 1 + G_ 0} \bigr) f + o(1) \] and uses this to get a central limit theorem for random evolutions driven by a continuous-time Markov chain.
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