Central limit theorem for stochastically continuous processes. Convergence to stable limit

Let X = {X(t), t is an element of [0, 1]} be a stochastically continuous cadlag process. Assume that the k dimensional finite joint distributions of X are in the domain of normal attraction of strictly p-stable, 0 < p < 2, measure on R(k) for all 1 less than or equal to k < infinity. For functions f, g such that Lambda(p)(\X(x)-X(u)\) < g(u-s) and Lambda(p)(\X(s)-X(t)\ boolean AND \X(t)-X(u)\) < f(u-s), 0 less than or equal to s less than or equal to t less than or equal to u less than or equal to 1, conditions are found which imply that the distributions L(n(-1/p)(X(1)+...+X(n))), n greater than or equal to 1, converge weakly in D[0, 1] to the distribution of a p-stable process. Here X(1), X(2),... are independent copies of X and Lambda(p)(Z) = sup(t > 0) t(p)P{\Z\ > t} denotes the weak pth moment of a random variable Z.