🚀 go-pugleaf

I have a problem that may fall into a standard class of problems delt with in statistical mechanics that utilizes large deviation theory.

My problem is to come up with an analytic (asymptotic) approximation to the following probability:

PR{ Sn < C}

where Sn= (Y1 + ... + Yn)/n

and Yi = min[0, Xi - Mi]

Mi are constants

X = (X1, ..., Xn)' are jointly normally distributed with differing variances but a constant correlation between any two elements of X.  

In short, I am interested in the asymptotic properties of sums of tails of correlated normal variates.

I expect that the solution has the form:

PR{Sn < C} === K exp(-nI) where both K and I are functions of the Mi's and the means and correlations of the X's.  Some times this expression has the more formal representation:

(1/n) log( PR{Sn < C} ) = I(parameters of the problem).

and I is refered to as a rate function.

If the X's were only iid there are results I've found that seem to apply.  The dependence structure appears to make this a much harder problem.

Any suggestions?