The Invariance Principle for Vector Valued Random Variables with Applications to Functional Random Limit Theorems

Abstract

1.Introduction. Let ξ₁, ξ₂,… be a sequence of independent real valued random variables with the same distribution with mean 0 and variance 1. Define the stochastic process ζ_n by

$$\zeta _n \left( t \right) = S_{\left[ {nt} \right]} /n^{1/2} = \left( {\xi _1 + \cdots \xi _{nt} } \right)/n^{1/2} ,$$

where 0⩽t⩽1 and n = 1, 2,… The classical Invariance Principle states that the sequence of D[0,1] valued random variables induced by the processes ζ_n(t) converges weakly to the Wiener process.