[R] Is the random number generator biased?

[Peter Dalgaard]

Oliver Faulhaber wrote:
> Hi all,
>
> in order to verify some results I did the following test in R (2.4.1., 
> windows system):
>
> X <- cumsum(rnorm(1000000))
> for (i in 1:1000) {
>   tmp                   <- seq(1,length(X),by=i)
>   X.coarse              <- X[tmp]
>   X.return              <- diff(X.coarse)
>   X.scale.mean[i]       <- mean(X.return)
> }
> plot(X.scale.mean,type="l")
>
> As X is a random walk with increments following a standard normal 
> distribution, the mean of X should be 0. Further more, each "piece" of 
> the random walk should also have zero mean - independent of its length.
>
> Why is it then, that the plot of X.scale.mean shows a clear linear trend?
>
> Is the generation of the random walk in some way biased or do I just 
> miss some point?
>
>   
The latter. If you think a little closer, you'll realize that 
sum(X.return) is going to be pretty close to X[1000000], except for the 
sum of at most 999 terms (at most 961, actually). You then proceed to 
calculate the mean by dividing by the number of terms which is 
essentially 1/i.

> Thanks for any enlighting
> replies in advance
> Oliver
>
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