If two random variables X and Y have the same first few moments, how different can their distributions be?
Suppose E[Xi] = E[Yi] for i = 0, 1, 2, … 2p. Then there is a polynomial P(x) of degree 2p such that
|F(x) – G(x)| ≤ 1/P(x)
where F and G are the CDFs of X and Y respectively.
The polynomial P(x) is given by
V‘ M-1 V
where V is a vector of dimension p+1 and M is a (p+1) × (p+1) matrix. The ith element of V is xi and the (i, j) element of M is E(Xi+j) if we start our indexes start from 0.
Reference: “Moments determine the tail of a distribution (but not much else)” by Bruce Lindsay and Prasanta Basak, The American Statistician, Vol 54, No 4, p. 248–251.