$\infty$-distributed

This definition of randomness is discussed in [8]. Let $U_1, U_2, U_3,\ldots$ denote a sequence of uniform random variates over [0,1). A sequence is $k$-distributed if

$$\mbox{Pr}(u_1 \le U_n < v_1, \ldots, u_k \le U_{n+k-1} < v_k) = (v_1-u_1) \ldots (v_k-u_k)$$

for all choices of real numbers $u_j, v_j$, with $0 \le u_j < v_j \le 1$, for all $1 \le j < k$. Put another way, each of $k$-dimensional points $(U_n, U_{n+1}, \ldots, U_{n+k-1})$, $n=1, 2, \ldots$ should occur the same number of times as $n$ approaches $\infty$. Note that when $k>1$, a $k$-distributed sequence is always a $(k-1)$-distributed sequence because we can set $u_k=0$ and $v_k=1$. A sequence is $\infty$-distributed (``super-uniform'') if it is $k$-distributed for any positive $k$. Obviously, this definition is only of theoretical interest and is not very useful in practice since computing resources like space and time are finite. However, within the confines of real-world, statistical tests like $\chi ^2$ test can measure how close a finite sequence is $k$-distributed for finite $k$.

This criterion is of central importance for stochastic simulations because all numbers in an $\infty$-distributed sequence are truly independent and have zero autocorrelation. It can also be proven that such a sequence can pass most if not all existing statistical tests.