Extract only a few coefficients of the multiple of extremely many polynomials

Question

I want to extract some informations, say the coefficients of $x^{500}$ and $x^{1000}$ terms (numerical values are acceptable), of the multiple of

(2/9 x^(-1) + 5/9 + 2/9 x)^700*(1/3 x^(-1) + 1/3 + 1/3 x)^600

However, the brutal computation would not be a practical way. I think the underlying idea is related to the convolution, but I can't see a way to do this. It would be good to have a lazy evaluation mechanism in MMA, but there seems no.

Actually this question comes from a probability task, that I want to compute the p.d.f. of the sum of, say 700 many random variables.

k = 700;
vec = Array[x, k];
dist = EmpiricalDistribution[{2/9, 5/9, 2/9} -> {-1, 0, 1}];
pdf = PDF@
  TransformedDistribution[Total[vec], 
   vec \[Distributed] ProductDistribution[{dist, k}]]
DiscretePlot[pdf[n], {n, -10, 10}, PlotRange -> All]
(*It took endless time... Hence not work.*)

How can I achieve this?

PS. If it is inevitable to neglects some intermediate terms with the norm of coefficients less than, say $2^{-300}$, to achieve the task, then I'm willing to do so. But how to do it?

Answer 1

Given

expr = (2/9 x^(-1) + 5/9 + 2/9 x)^700*(1/3 x^(-1) + 1/3 + 1/3 x)^600;

The coefficient of $x^{500}$ can be computed as an exact rational in about a minute or so

Coefficient[expr, x, 500]
(* output omitted *)

If you just need a numerical value, the following is faster (and agrees with the rational value)

Coefficient[Map[N[#, 20] &, expr, {3}], x, 500]
(* 3.2222219661144821*10^-81 *)

As @Somos suggests, if your real purpose is understanding the statistical distribution, applying the central limit theorem is probably likely to be helpful. Of course, probabilities of $10^{-81}$ are probably not practically meaningful.