Hyperplane problem in linear congruent generator From Wikipedia

if an LCG is used to choose points in an n-dimensional space, the
points will lie on, at most, $m^{1/n}$ hyperplanes (Marsaglia's Theorem,
developed by George Marsaglia). This is due to serial correlation
between successive values of the sequence $X_n$.



*

*Mathematically, what is the serial correlation between successive values of the
sequence $X_n$?


*I was wondering how the
points within one period are distributed to different hyperplanes, as they are generated one after another in a sequence?  I guess it is unlikely that the points firstly fill out one hyperplane,
and then fill out the next one, and will never visit the hyperplanes
visited before in the single period?
Is the time interval between every two consecutive visits to each fixed hyperplane fixed, and same for all hyperplanes? Is it connected to the period of the LCG?
Does taking a small front portion of a sequence of a full period somehow overcome this drawback of LCG? That is the reason why I ask the above questions.


*Note I think there is a typo. "at most, $m^{1/n}$ hyperplanes"
should be "at most, $(n!m)^{1/n}$ hyperplanes". Am I correct?
Thanks!
 A: *

*$\text{corr}(R_t,R_{t-1}) \leq \frac{1}{a}\left(1-\frac{6c}{m}+6(\frac{_{c}}{^m})^2\right)+\frac{a+6}{m}$
This formula is an adaptation from Greenberger, 1961$^{[1]}$, who gives the first term above in an initial approximation and later gives another approximation with an additional term.
The serial correlation is typically quite small, even for a poor generator

*your own animation indicates that the points don't "firstly fill out one hyperplane", since you see different planes get more points over time. The first step of the animation shows points from several different planes, then more, and as it progresses, the earlier planes get more points

Is the time interval between every two consecutive visits to each fixed hyperplane fixed, and same for all hyperplanes? 

This I don't know for certain. I believe there's a cycle through the hyperplanes.

Does taking a small front portion of a sequence of a full period somehow overcome this drawback of LCG?

No.

*Yes, it should. An edit to the indicated page as at 31 May 2014 has made it match your formula
I'd suggest looking at the DIEHARD tests and similar tests of RNGs
$\ $
[1]: Greenberger, Martin  (1961),
 "An a priori determination of serial correlation in computer generated random numbers",
Math. Comp. 15, 383-389    (pdf)
 (Corrigendum: Math. Comp. 16 (1962), 406-406.)  
