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Hi everyone,

I was wondering is there were some known applications of MCMC methods for simulating the law of the sum ( or alternatively the joint law) of the order statistics of a multinomial (centered) gaussian vector with given full rank covariation matrix of large dimension.

For example let's be given such a vector $N\to \mathcal{N}(0,\Gamma)$ with $\Gamma$ a full rank matrix of dimenson $P$.

If I want to get the law of the sum of the $k< P $ first order statisitics $S_{(k)}=\sum_{i=1}^k N_{(i)}$, I can simulate $N$ via Monte Carlo but the thing is that it is very time consumming as I have to simulate the $P$ components of the vector $N$.

My Question : Is there a way in this specific case to get a Markov Chain of dimension $k$ that can be shown to converge to the stationary distribution of the joint law of $S_{(k)}$ ?

So I can get the law of the sum.

Or alternatively (even better) is there a way to find a chain of dimension 1 converging directly to the law of the sum of the first $k$ order statistics ?

Then using this "way", Ithink that I can gain some huge amount of computation time vis-à-vis the naive Monte Carlo method.

Best Regards

PS: As far as I know there is no analytcal formula for the law of $S_{(k)}$ for general gaussian vector.

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    $\begingroup$ What is the dimension p that causes you so much trouble for generating a guassian random variable ? $\endgroup$
    – robin girard
    Commented Jul 27, 2010 at 15:17
  • $\begingroup$ Hi Robin, The number $P$ is something like 40,and k around 5, I have no trouble generating a 40-dim Gaussian vector but I wanted to check is there was a faster way to do thing. I am as a matter of fact considering the use ACP first and then make low dimensional approximation of $S_{(k)}$, but as MCMC seems to be the "big thing" right now I was just curious to know if this method was applicable here with succes (i.e. with a fast convergence) Regards $\endgroup$
    – The Bridge
    Commented Jul 27, 2010 at 15:56
  • $\begingroup$ @Bridge with p=40 you can simulate everything easely using R, and you won't need MCMC. $\endgroup$
    – robin girard
    Commented Jul 27, 2010 at 17:44
  • $\begingroup$ You are right Robin, I just wanted to "challenge" the method in a practical case where I know how to sample the distribution in a straithforward way. $\endgroup$
    – The Bridge
    Commented Aug 3, 2010 at 13:45

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