• DOI:10.1090/S0002-9947-1957-0084222-7
• Corpus ID: 53557322

ON OCCUPATION TIMES FOR MARKOFF PROCESSES

@article{Darling1957ONOT,
  title={ON OCCUPATION TIMES FOR MARKOFF PROCESSES},
  author={Donald Darling and Mark Kac},
  journal={Transactions of the American Mathematical Society},
  year={1957},
  volume={84},
  pages={444-458}
}

• D. Darling, M. Kac
• Published 1 February 1957
• Mathematics
• Transactions of the American Mathematical Society

where u(t) is a suitable normalization. If V(x) is the characteristic function of a set, ftaV(x(r))dT is the occupation time of the set. The principal result is that under suitable (but quite general) conditions the limiting distribution must be the Mittag-Leffler distribution (of an appropriate index). The method of proof is equally applicable to Markoff chains and, in particular, to sums of independent, identically distributed random variables. We thus obtain a considerable generalization and unification of previous results of Feller [l], Chung and Kac [2] and Kallianpur and Robbins [3; 4]. It will be seen that the somewhat lengthy computations of these authors can be dispensed with by virtue of the elementary Tauberian theorem of Karamata. Finally the distribution of the number of changes of sign in a sequence of partial sums of identically distributed random variables will also emerge as an application of our general theory. 2. A special cese. In order to illustrate the method and bring out clearly the role of assumptions under which the general theorem will be proved we shall first consider a special case. Let x(t), t^O be the two dimensional Brownian motion, x(0) =0, and let V(x) he the characteristic function of a bounded plane set B of nonzero Lebesgue measure. Let us calculate the moments of f0V(x(T))dr. Consider e.g. the second moment