The central limit theorem for empirical and quantile processes in some Banach spaces (Q1802316)

Let \(\alpha_ n(t)\), \(0<t<1\), \(n\in N\), be a sequence of empirical processes constructed by means of a sequence of independent uniformly distributed on \((0,1)\) random variables. The sequence of finite- dimensional distributions of \((\alpha_ n)\) converges weakly to those of Brownian bridge \(B(t)\), \(0\leq t\leq 1\). Let also \(X\) be a Banach function space on \((0,1)\) such that the realizations of \(B\) and \(\alpha_ n\), \(n\in N\), belong to \(X\) a.s. Then, for which \(X\) we can say that the sequence of distributions of \(\alpha_ n\), \(n\in N\), in \(X\) converges weakly to the distribution of \(B\) in \(X\)? This question is answered affirmatively when \(X\) is the Orlicz space \(L_ \Phi\), where \(\Phi\) satisfies the \(\Delta_ 2\)-condition. This is derived from obvious results concerning the CLT in Banach spaces.