SOME LOCAL RESULTS ON THE CONVERGENCE AND THE QUADRATIC VARIATION OF RANDOM SEQUENCES

In this paper we study the following property {Mathematical expression} of a random sequence (Xn, n∈ℕ):Xn converges a.s. to a finite limit and its quadratic variation is finite a.s. Several local results on the property {Mathematical expression} are established for random sequences satisfying {Mathematical expression} where (hn, n∈ℕ) is a random sequence. As an application random sequences taking values in a closed, possibly unbounded, interval of- are studied. © 1990 Springer-Verlag.