Central limit theorem and increment conditions
Suppose that \((X_ t,0\leq t\leq 1)\) is a uniformly bounded separable process. A remarkable recent result of \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037); Theorem 7.4] shows that if for every \(u<t\), \(| X_ t-X_ n| \leq t-u\), then \((X_ t)\) satisfies the central limit theorem. In the present paper it is shown that if \(\phi\) is a convex function defined on \(R^+\) with \(\phi (0)=0\) and \(\lim_{x\to 0}\phi (x)/x=0\), then there exists a uniformly bounded separable process \((X_ t,0\leq t\leq 1)\) with continuous sample paths such that for every \(u<t\) \(E[\phi (| X_ t-X_ u|)]\leq t-u\) but for which the central limit theorem does not hold. Thus, the Giné and Zinn result is close to optimal.
scientific article; zbMATH DE number 4007336