A note on the central limit theorem for stochastically continuous processes

The desymmetrization technique which was successfully used in C(S) spaces is carried over to limit theorems for stochastically continuously random processes with sample paths in Skorohod space D[0, 1] and is applied to obtain the central limit theorem (CLT) in D[0, 1]. Let {X(t), t [set membership, variant] [0, 1]} be a stochastically continuous random process. For functions [latin small letter f with hook], g such that E(X(s) - X(t) [logical and] X(t) - X(u))p <= [latin small letter f with hook](u - s), EX(s) - X(t)q <= g(t - s), p, q >= 2, s <= t <= u, conditions are found which imply the CLT in D[0, 1].