A Short Proof for Hausdorff Moment Problem

Hausdorff moment problem asks for necessary and sufficient conditions that a given sequence (m_n) with m_0=1 be the sequence of moments of a random variable X supported on [0,1], i.e., \operatorname{E}X^n=m_n for all n.

In 1921, Hausdorff showed that (m_n) is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation (D^r m)_s \ge 0 for all r, s \ge 0. Here D is the difference operator on the space of real sequences (a_n) given by D a = (a_{n} - a_{n+1}).

The proof under the fold follows the outline given in (E18.5 – E18.6) Probability with Martingales by David Williams.

Proof of Necessity. Suppose (m_n) is the moment sequence of a random variable X supported on [0,1]. By induction, one can show that (D^r m)_s = \operatorname{E}(1-X)^rX^s. Clearly, as X is supported on [0,1], the moment sequence is completely monotonic.

Proof of Sufficiency. Suppose (m_n) is a completely monotonic sequence with m_0 = 1.

Define F_n(x) := \sum_{i \le nx}{n\choose i}(D^{n-i}m)_i. Clearly, F_n is right-continuous and non-decreasing, and F_n(0^-) = 0. To prove F_n(1) = 1, one has to prove the identity \sum_{i}{n\choose i}(D^{n-i}m)_i = m_0.

A classical trick. Since the identity above is about vectors in the linear space (over the reals) spanned by (m_n) and the linear space spanned by (m_n) is isomorphic to the one spanned by (\theta^n), the identity is equivalent to \sum_{i}{n\choose i}(D^{n-i}\theta)_i = \theta^0, where \theta_n = \theta^n. Now, we take advantage of the ring structure of \mathbb{R}[\theta]. Notice that (D^{r}\theta)_s = (1-\theta)^r\theta^s. Using the binomial theorem, we obtain \sum_{i}{n\choose i}(D^{n-i}\theta)_i = \sum_{i}{n\choose i}(1-\theta)^{n-i}\theta^i = (1-\theta + \theta)^n = \theta^0.

Therefore F_n is a bona fide distribution function. Define m_{n, k} := \int_{[0,1]} x^kdF_n(x), i.e., m_{n,k} is the kth moment of F_n. We now find an explicit formula for m_{n,k}.

Noticing that F_n is constant, say c_{n,i}, on [\frac{i}{n}, \frac{i+1}{n}), for all i=0, \dots, n-1 and c_{n,i} is a linear combination of m_0, \dots, m_n, we know that m_{n,k} = \sum_{i=0}^n a_{n,k,i}m_i.

Just like what we did in proving the identity, we use the special case m_n = \theta^n to compute the coefficients a_i = a_{n,k,i}, where 0 \le \theta \le 1. In this case, F_n(x) = \sum_{i \le nx}{n\choose i}(D^{n-i}\theta)_i = \sum_{i\le nx}{n\choose i}(1-\theta)^{n-i}\theta^i, m_{n,k} = \sum_{i=0}^n a_{i}\theta^i.

Now consider the situation in which a coin with probability \theta is tossed at times 1,2,\dots. The random variable H_k is 1 if the kth toss produces heads, 0 otherwise. Define A_n := (H_1 + \dots + H_n)/n. It is immediate from the formula of F_n that F_n is the distribution function of A_n, and so m_{n,k} is the kth moment of A_n. However, one can calculate the kth moment of A_n explicitly. Let f\colon [k] \to [n] be chosen uniformly at random, Im_f be the cardinality of the image of f and denote by p_i = p_{n,k,i} := \operatorname{Pr}(Im_f = i). Using f, Im_f and p_i, we obtain \begin{aligned}\operatorname{E}A_n^k & = \operatorname{E}\left(\frac{H_1 + \dots + H_n}{n}\right)^k = \operatorname{E}H_{f(1)}\dots H_{f(k)} \\ & = \operatorname{E}\operatorname{E}[H_{f(1)}\dots H_{f(k)}\mid Im_f] = \operatorname{E}\theta^{Im_f} = \sum_{i=0}^n p_{i}\theta^i.\end{aligned} Therefore, for all \theta\in [0,1], we know that \sum_{i=0}^n a_i\theta^i = \sum_{i=0}^n p_i\theta^i, and so a_i = p_i for all i=0,\dots, n.

As both (a_i) and (p_i) do not depend on m_i, a_i = p_i holds in general. Since p_k = p_{n, k, k} = \prod_{i=0}^{k-1}(1-i/n)\to 1 as n\to\infty and p_i = 0 for all i > k, we know that \lim_n m_{n,k}= m_k.

Using the Helly–Bray Theorem, since (F_n) is tight, there exists a distribution function F and a subsequence (F_{k_n}) such that F_{k_n} converges weakly to F. The definition of weak convergence implies that \int_{[0,1]} x^k dF(x) = \lim_n \int_{[0,1]}x^k dF_{k_n}(x) = \lim_n m_{k_n,k} = m_k. Therefore, the random variable X with distribution function F is supported on [0,1] and its kth moment is m_k.

There are other two classical moment problems: the Hamburger moment problem and the Stieltjes moment problem.