Central limit theorem in \(D[0,1]\) (Q2726255)

Let \(D\equiv D[0,1]\) denote the space of real-valued functions on \([0,1]\) which are right-continuous on \([0,1]\) with left limits on \((0,1],\) which is endowed with the Skorokhod topology. The background for the theory of the weak convergence of stochastic processes in \(D\) was laid in papers by \textit{Yu. V. Prokhorov} [Teor. Veroyatn. Primen. 1, 177-238 (1956; Zbl 0075.29001)], \textit{A. V. Skorokhod} [ibid. 1, 289-319 (1956; Zbl 0074.33802)] and \textit{N. N. Chentsov} [Dokl. Akad. Nauk SSSR 106, 607-609 (1956; Zbl 0074.12503)]. Later on the book by \textit{P. Billingsley} [``Convergence of probability measures'' (1968; Zbl 0172.21201)] became the most popular reference book for the weak convergence of measures on metric spaces, in particular on the space \(D.\) NEWLINENEWLINENEWLINEThis paper surveys recent results on the central limit theorem for stochastically continuous processes having sample paths in the Skorokhod space \(D.\) The refined tightness criteria in the space \(D\) are considered which are applied to establish sufficient conditions for the central limit theorem (CLT) in \(D.\) Two applications of the CLT in \(D\) are given: the weak convergence of weighted empirical processes and the asymptotic distribution of the fiber bundle strength in the fiber bundle model introduced by \textit{H. E. Daniels} [Proc. Roy. Soc. London, Ser. A 183, 405-435 (1945)]. An estimate of the rate of convergence in the CLT in \(D\) is presented. The presented results are extended to the space \(D_{k}\) (the Skorokhod space of \(k\)-variate real càdlàg functions on \([0,1]^{k}\)).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].