Suppose we wanted to choose random locations in the unit square for Monte Carlo
trials in an application program. A very easy way to choose these points is to
select two random values, 
and 
, in [0,1) and
choosing the point 
as the current point of
interest in the square. More generally, we
would generate a point by plotting 
vs. 
, where 
and 
are of course obtained by scaling successive outputs, 
and
, of the generator 
by 
. If we repeat this
procedure for a large number of trials, we would like to expect that we will
achieve a reasonably good ``covering'' of the unit square and in a ``randomly''
ordered fashion. We would be suspicious of a generator that produced points in
the square that were, say, clustered in the bottom half, or perhaps covered the
square in some clear order from left to right. Many other forms of obvious
nonrandom behavior would be equally unacceptable for most Monte Carlo
applications. A characteristic of LCGs is that points selected in this way and
plotted in the unit square begin to form regular-looking rows or dotted lines
that are easily discernible when enough points have been plotted and
when viewed
at the proper scale. Over the entire period of an LCG, if all
consecutive pairs
are plotted, then these rows fill in to become evenly spaced between points.
