# Non Diagonal Case

Chapter
Part of the Lecture Notes in Statistics book series (LNS, volume 71)

## Abstract

Chapter 4 provided a detailed study of the behavior of the random vector (tX)(mod 1) for large values of t. From the fact that
$$tX = \left( {\begin{array}{*{20}c} {tX_1 } \\ {tX_2 } \\ \vdots \\ {tX_n } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} t \\ 0 \\ \vdots \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ t \\ \vdots \\ 0 \\ \end{array} \begin{array}{*{20}c} \cdots \\ \cdots \\ \ddots \\ \cdots \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ \vdots \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {X_1 } \\ {X_2 } \\ \vdots \\ {X_n } \\ \end{array} } \right),$$
this may be viewed as the “diagonal case.” This chapter deals with the general case, that is, given an n dimensional random vector X and a collection of n by n matrices, A(τ); τT ⊂ IR, attention is focused on conditions under which (A(τ)X)(mod 1) converges to a distribution uniform on the unit hypercube, Un, as τ tends to infinity.