Next: Unpredictable
Up: Randomness Assessments
Previous: Randomness Assessments
-distributed
This definition of randomness is discussed in [8].
Let
denote a sequence of uniform random variates over [0,1).
A sequence is
-distributed if
for all choices of real numbers
, with
, for
all
.
Put another way, each of
-dimensional points
,
should occur the same number of times as
approaches
.
Note that when
, a
-distributed sequence is always a
-distributed sequence
because we can set
and
. A sequence is
-distributed (``super-uniform'')
if it is
-distributed for any positive
. 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
test can measure how close a finite sequence is
-distributed
for finite
.
This criterion is of central importance for stochastic simulations because all numbers
in an
-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.
Next: Unpredictable
Up: Randomness Assessments
Previous: Randomness Assessments
2001-05-30