Extension of Donskers Invariance Principle with Incomplete Partial-Sum Process

Based on deleting-item central limit theory, the classical Donskers theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete partial-sum process Donskers invariance principles are constructed and derived for general partial-sum process of i.i.d random variables and empirical process respectively, they are not only the extension of functional central limit theory, but also the extension of deleting-item central limit theory. Our work enriches the random elements structure of weak convergence.