author={Donald Darling and Mark Kac},
  journal={Transactions of the American Mathematical Society},
  • D. DarlingM. 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… 


time (properly normalized) are the Mittag-Leffler distributions, while in the case of a process consisting of sums of independent, identically-distributed random variables, limiting distributions of

Limit Theorems of Occupation Times for Markov Processes

where u(f) is some normalizing function, has been investigated by many authors. A most general limit theorem was obtained by Darling and Kac [1], who also showed that, under suitable condition, the

A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case

  • P. S. Puri
  • Mathematics
    Journal of Applied Probability
  • 1971
The subject of this paper is the study of the distribution of integrals of the type where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable

Limit theorems for some continuous-time random walks

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {X n , n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and

Some Theorems on Functionals of Markov Chains

1. Introduction. In this paper we shall investigate various phenomena associated with a Markov process in discrete time, extending results found in [3], [6], [7], and [13]. The paper is divided into

Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes.

The correlator method is extended to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings.

First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the

Non-markovian limits of additive functionals of Markov processes

In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process. We assume that the law of the expected jump time $t(x)$ under the invariant probability measure

Random walks and a sojourn density process of Brownian motion

The sojourn times for the Brownian motion process in 1 dimension have often been investigated by considering the distribution of the length of time spent in a fixed set during a fixed time interval.



The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$

where L consists of three parts as follows: &: the line y= —(tan \f/)x from x— + °° to # = p , p > 0 . C2: an arc of circle \z\ = p sec \p> — ̂ ^ a r g z^\//. Cz\ the reflection of G in the x-axis.

Fluctuation theory of recurrent events

The problems treated in this paper belong to pure analysis, but they have an intuitive background which can be roughly described as follows. We are concerned with sequences of repeated trials which

Ergodic Property of the Brownian Motion Process.

  • C. Derman
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1954

Dobrusin, Two limit theorems for the simplest random walk on a line, Uspehi Matematifiskih

  • Nauk (N.S.) vol
  • 1955