The central limit theorem in the space \(D[0,1]\). I
Let \(D=D[0,1]\) denote the space of real-valued functions on \([0,1]\), which are right continuous and possess left-hand limits with the Skorokhod topology. The main result of the present paper is as follows. Let \(X = \{X(t), t \in [0,1]\}\) be a random process with finite second moments and \(EX(t) = 0\) for all \(t \in [0,1]\). Let \(f\) and \(g\) be nonnegative and nondecreasing functions defined on \([0, \infty)\). If for all \(0 \leq s \leq t \leq u \leq 1\), \[ E (X(u) - X(s))^2 \leq g(u - s),\quad E (|X(t) - X(s) |\wedge 1)^2\;(X(u) - X(s))^2 \leq f(u - s), \] \[ \int^1_0 y^{- 5/4}f^{1/4} (y)dy < \infty,\quad \int^1_0 y^{- 9/8} g^{1/4} (y) dy < \infty, \] then there exists a version \(X'\) of the random process \(X\) with sample paths in \(D[0,1]\) and \(X'\) satisfies the central limit theorem in the space \(D\). [For part II see below].
scientific article; zbMATH DE number 794486