Wall's continued-fraction characterization of Hausdorff moment sequences: A conceptual proof

TL;DR

This paper provides an elementary proof of Wall's characterization of Hausdorff moment sequences using continued fractions, simplifying the understanding of their mathematical structure.

Contribution

The paper offers a new, elementary proof of Wall's continued-fraction characterization, making the concept more accessible and easier to understand.

Findings

Elementary proof of Wall's characterization

Simplifies understanding of Hausdorff moment sequences

Clarifies the structure using continued fractions

Abstract

I give an elementary proof of Wall's continued-fraction characterization of Hausdorff moment sequences.

Equations

$$\sum _{n=0}^{\infty }{a}_n{t}^n=\frac{{\alpha }_0}{1-\frac{{\alpha }_{1}t}{1-\frac{{\alpha }_{2}t}{1-\cdots }}}$$

$$\sum _{n=0}^{\infty }{a}_n{t}^n=\frac{c}{1-\frac{g_{1}t}{1-\frac{(1-g_1)g_{2}t}{1-\frac{(1-g_2)g_{3}t}{1-\frac{(1-g_3)g_{4}t}{1-\cdots }}}}}$$

$$f(t)=\frac{1}{1-{\gamma }_{0}t-\frac{{\beta }_1{t}^2}{1-{\gamma }_{1}t-\frac{{\beta }_2{t}^2}{1-\cdots }}}$$

$$\frac{1}{1-\frac{{\alpha }_{1}t}{1-\frac{{\alpha }_{2}t}{1-\frac{{\alpha }_{3}t}{1-\cdots }}}}=\frac{1}{1-{\alpha }_{1}t-\frac{{\alpha }_1{\alpha }_2{t}^2}{1-({\alpha }_2+{\alpha }_3)t-\frac{{\alpha }_3{\alpha }_4{t}^2}{1-({\alpha }_4+{\alpha }_5)t-\frac{{\alpha }_5{\alpha }_6{t}^2}{1-\cdots }}}}$$

$$\gamma_{0}&=\alpha_{1}{}\\ \gamma_{n}=\alpha_{2n}+\alpha_{2n+1}\quad\hbox{for $n\geq 1$}{}\\ \beta_{n}=\alpha_{2n-1}\alpha_{2n}{}$$

$$b_n=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k{\xi }^{n-k}.$$

$$\sum _{n=0}^{\infty }{a}_n{t}^n=\frac{1}{1-{\gamma }_{0}t-\frac{{\beta }_1{t}^2}{1-{\gamma }_{1}t-\frac{{\beta }_2{t}^2}{1-\cdots }}}.$$

$$\sum _{n=0}^{\infty }{b}_n{t}^n=\frac{1}{1-({\gamma }_0+\xi )t-\frac{{\beta }_1{t}^2}{1-({\gamma }_1+\xi )t-\frac{{\beta }_2{t}^2}{1-\cdots }}}.$$

$$\sum _{n=0}^{\infty }{\hat{a}}_n{t}^n=\sum _{n=0}^{\infty }{a}_n{t}^{2n}=\frac{1}{1-\frac{{\alpha }_1{t}^2}{1-\frac{{\alpha }_2{t}^2}{1-\cdots }}}.$$

$$\sum _{n=0}^{\infty }{\tilde{a}}_n{t}^n=\frac{1}{1-t-\frac{{\alpha }_1{t}^2}{1-t-\frac{{\alpha }_2{t}^2}{1-\cdots }}}.$$

$$&1\,=\,\alpha^{\prime}_{1}\,=\,\alpha^{\prime}_{2}+\alpha^{\prime}_{3}\,=\,\alpha^{\prime}_{4}+\alpha^{\prime}_{5}\,=\,\ldots{}\\ \alpha_{1}\,=\,\alpha^{\prime}_{1}\alpha^{\prime}_{2}\,,\quad\alpha_{2}\,=\,\alpha^{\prime}_{3}\alpha^{\prime}_{4}\,,\quad\ldots$$

Full text

Key Words: Classical moment problem, Hamburger moment sequence, Stieltjes moment sequence, Hausdorff moment sequence, binomial transform, continued fraction.

Mathematics Subject Classification (MSC 2010) codes: 44A60 (Primary); 05A15, 30B70, 30E05 (Secondary).

Let us recall that a sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ of real numbers is called a Hamburger (resp. Stieltjes, resp. Hausdorff) moment sequence [19, 1, 3, 20, 17] if there exists a positive measure $\mu$ on ${\mathbb{R}}$ (resp. on $[0,\infty)$, resp. on $[0,1]$) such that $a_{n}=\int\!x^{n}\,d\mu(x)$ for all $n\geq 0$. One fundamental characterization of Stieltjes moment sequences was found by Stieltjes [22] in 1894 (see also [26, pp. 327–329]): A sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ of real numbers is a Stieltjes moment sequence if and only if there exist real numbers $\alpha_{0},\alpha_{1},\alpha_{2},\ldots\geq 0$ such that

[TABLE]

in the sense of formal power series. (That is, the ordinary generating function $f(t)=\sum\limits_{n=0}^{\infty}a_{n}t^{n}$ can be represented as a Stieltjes-type continued fraction with nonnegative coefficients.) Moreover, the coefficients ${\bm{\alpha}}=(\alpha_{i})_{i\geq 0}$ are unique if we make the convention that $\alpha_{i}=0$ implies $\alpha_{j}=0$ for all $j>i$; we shall call such a sequence ${\bm{\alpha}}$ standard.

Since every Hausdorff moment sequence is a Stieltjes moment sequence, its ordinary generating function clearly has a continued-fraction expansion of the form (1) with coefficients ${\bm{\alpha}}\geq 0$. But which sequences ${\bm{\alpha}}\geq 0$ correspond to Hausdorff moment sequences? The answer was given by Wall [24, Theorems 4.1 and 6.1] in 1940: A sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ of real numbers is a Hausdorff moment sequence if and only if there exist real numbers $c\geq 0$ and $g_{1},g_{2},g_{3},\ldots\in[0,1]$ such that

[TABLE]

in the sense of formal power series.

Wall’s proof of this result was based on an interesting but somewhat mysterious identity for continued fractions [24, Theorem 2.1] together with some complex-analysis arguments.111 See also [12] for a combinatorial proof of Wall’s identity, and see [15, 27, 9] for some interesting applications of it.

Four years later, Wall [25] gave a new proof, based on Schur’s [18] characterization of analytic functions bounded in the unit disc and the Herglotz–Riesz [10, 16] integral representation of analytic functions in the unit disc with positive real part.

Here I would like to present an alternate proof of Wall’s theorem that is not only very simple but also gives insight into why the coefficients in (2) take the form $\alpha_{n}=(1-g_{n-1})g_{n}$.

This proof requires two well-known elementary facts about moment sequences:

1. ${\bm{a}}$ is a Stieltjes moment sequence if and only if the “aerated” sequence $\widehat{{\bm{a}}}=(a_{0},0,a_{1},0,a_{2},0,\ldots)$ is a Hamburger moment sequence. Indeed, the even subsequence of a Hamburger moment sequence is always a Stieltjes moment sequence; and conversely, if ${\bm{a}}$ is a Stieltjes moment sequence that is represented by a measure $\mu$ supported on $[0,\infty)$, then $\widehat{{\bm{a}}}$ is represented by the even measure $\widehat{\mu}=(\tau^{+}+\tau^{-})/2$ on ${\mathbb{R}}$, where $\tau^{\pm}$ is the image of $\mu$ under the map $x\mapsto\pm\sqrt{x}$. In particular, if $\mu$ is supported on $[0,\Lambda]$, then $\widehat{\mu}$ is supported on $[-\sqrt{\Lambda},\sqrt{\Lambda}]$.

2. If the Hamburger moment sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ satisfies $|a_{n}|\leq AB^{n}$ with $A,B<\infty$, then the representing measure $\mu$ is unique and is supported on $[-B,B]$. In particular, a Hausdorff moment sequence always has a unique representing measure. (In fact, the representing measure $\mu$ of a Hamburger moment sequence is unique under the vastly weaker hypothesis $|a_{n}|\leq AB^{n}n!$, or even under the yet weaker hypothesis $\sum\limits_{n=1}^{\infty}a_{2n}^{-1/2n}=\infty$ [17, section 4.2]; but we shall not need these latter results.)

Besides Stieltjes-type continued fractions (1) [henceforth called S-fractions for short], we shall also make use of Jacobi-type continued fractions (J-fractions)

[TABLE]

(always considered as formal power series in the indeterminate $t$).222 My use of the terms “S-fraction” and “J-fraction” follows the general practice in the combinatorial literature, starting with Flajolet [8]. The classical literature on continued fractions [14, 26, 11, 13, 4] generally uses a different terminology. For instance, Jones and Thron [11, pp. 128–129, 386–389] use the term “regular C-fraction” for (a minor variant of) what I have called an S-fraction, and the term “associated continued fraction” for (a minor variant of) what I have called a J-fraction.

We shall need three elementary facts about these continued fractions:

1. The contraction formula: We have

[TABLE]

as an identity between formal power series. In other words, an S-fraction with coefficients ${\bm{\alpha}}$ is equal to a J-fraction with coefficients ${\bm{\gamma}}$ and ${\bm{\beta}}$, where

[TABLE]

See [26, pp. 20–22] for the classic algebraic proof of the contraction formula (4); see [7, Lemmas 1 and 2] [6, proof of Lemma 1] [5, Lemma 4.5] for a very simple variant algebraic proof; and see [23, pp. V-31–V-32] for an enlightening combinatorial proof, based on Flajolet’s [8] combinatorial interpretation of S-fractions (resp. J-fractions) as generating functions for Dyck (resp. Motzkin) paths with height-dependent weights.

2. Binomial transform: Fix a real number $\xi$, and let ${\bm{a}}=(a_{n})_{n\geq 0}$ be a sequence of real numbers. Then the $\xi$-binomial transform of ${\bm{a}}$ is defined to be the sequence ${\bm{b}}=(b_{n})_{n\geq 0}$ given by

[TABLE]

Note that if $a_{n}=\int\!x^{n}\,d\mu(x)$, then $b_{n}=\int(x+\xi)^{n}\,d\mu(x)$. In other words, if ${\bm{a}}$ is a Hamburger moment sequence with representing measure $\mu$, then ${\bm{b}}$ is a Hamburger moment sequence with representing measure $T_{\xi}\mu$ (the $\xi$-translate of $\mu$).

Now suppose that the ordinary generating function of ${\bm{a}}$ is given by a J-fraction:

[TABLE]

Then the $\xi$-binomial transform ${\bm{b}}$ of ${\bm{a}}$ is given by a J-fraction in which we make the replacement $\gamma_{i}\to\gamma_{i}+\xi$:

[TABLE]

See [2, Proposition 4] for an algebraic proof of (8); or see [21] for a simple combinatorial proof based on Flajolet’s [8] theory.

3. An upper bound: If ${\bm{a}}$ is given by the S-fraction (1) with $0\leq\alpha_{i}\leq 1$ for all $i$, then $0\leq a_{n}\leq C_{n}\leq 4^{n}$, where $C_{n}=\displaystyle\frac{1}{n+1}\binom{2n}{n}$ is the $n$th Catalan number. The proof is simple: If we consider the coefficients ${\bm{\alpha}}$ in (1) to be indeterminates, then it is easy to see that $a_{n}$ is a polynomial in $\alpha_{0},\ldots,\alpha_{n}$ with nonnegative integer coefficients (namely, $\alpha_{0}$ times the Stieltjes–Rogers polynomial $S_{n}({\bm{\alpha}})$ [8]); so $a_{n}$ is an increasing function of ${\bm{\alpha}}$ on the set ${\bm{\alpha}}\geq 0$. On the other hand, if $\alpha_{i}=1$ for all $i$, then (1) represents a series $f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}$ satisfying $f(t)=1/[1-tf(t)]$, from which it follows that $f(t)=[1-\sqrt{1-4t}]/(2t)$ and hence (by binomial expansion) that $a_{n}=C_{n}$.

Proof of Wall’s theorem.  Let ${\bm{a}}=(a_{n})_{n\geq 0}$ be a Hausdorff moment sequence; we can assume without loss of generality that $a_{0}=1$. Then ${\bm{a}}$ has a (unique) representing measure $\mu$ supported on $[0,1]$, and its ordinary generating function is given by a unique S-fraction (1) with $\alpha_{0}=1$ and standard coefficients ${\bm{\alpha}}\geq 0$. Now let $\widehat{{\bm{a}}}=(a_{0},0,a_{1},0,a_{2},0,\ldots)$ be the aerated sequence; it is a Hamburger moment sequence with a (unique) even representing measure $\widehat{\mu}$ supported on $[-1,1]$, and its ordinary generating function is given by the J-fraction with coefficients ${\bm{\gamma}}=0$ and ${\bm{\beta}}={\bm{\alpha}}$:

[TABLE]

Now let $\widetilde{{\bm{a}}}$ be the 1-binomial transform of $\widehat{{\bm{a}}}$; it is a Stieltjes moment sequence with a (unique) representing measure $\widetilde{\mu}=T_{1}\widehat{\mu}$ supported on $[0,2]$, and its ordinary generating function is given by a J-fraction with coefficients ${\bm{\gamma}}={\bm{1}}$ and ${\bm{\beta}}={\bm{\alpha}}$:

[TABLE]

But since $\widetilde{{\bm{a}}}$ is a Stieltjes moment sequence, its ordinary generating function is also given by an S-fraction with nonnegative coefficients, call them ${\bm{\alpha}}^{\prime}$. Comparing the J-fraction and the S-fraction using the contraction formula (4)/(5), we see that

[TABLE]

It follows from (LABEL:eq.alphaprime.contraction.a) that $\alpha^{\prime}_{i}\in[0,1]$ for all $i$. Setting $g_{n}=\alpha^{\prime}_{2n}$ shows that $\alpha_{1}=g_{1}$ and $\alpha_{n}=(1-g_{n-1})g_{n}$ for $n\geq 2$, which is precisely the representation (2).

Conversely, suppose that ${\bm{a}}$ is given by an S-fraction (2) with coefficients $c=1$ and $g_{i}\in[0,1]$. Then ${\bm{a}}$ is a Stieltjes moment sequence satisfying $a_{n}\leq 4^{n}$, so that the representing measure $\mu$ is unique and is supported on $[0,4]$. (Of course, we will soon see that $\mu$ is actually supported on $[0,1]$.) Then the aerated sequence $\widehat{{\bm{a}}}=(a_{0},0,a_{1},0,a_{2},0,\ldots)$ is a Hamburger moment sequence with a unique representing measure $\widehat{\mu}$ that is even and supported on $[-2,2]$, and its ordinary generating function is given by a J-fraction with coefficients ${\bm{\gamma}}=0$, $\beta_{1}=g_{1}$ and $\beta_{n}=(1-g_{n-1})g_{n}$ for $n\geq 2$. Now let $\widetilde{{\bm{a}}}$ be the 1-binomial transform of $\widehat{{\bm{a}}}$: it is a Hamburger moment sequence with a unique representing measure $\widetilde{\mu}=T_{1}\widehat{\mu}$ supported on $[-1,3]$, and its ordinary generating function is given by a J-fraction with coefficients ${\bm{\gamma}}={\bm{1}}$, $\beta_{1}=g_{1}$ and $\beta_{n}=(1-g_{n-1})g_{n}$ for $n\geq 2$. But the contraction formula (4)/(5) shows that this J-fraction is equivalent to an S-fraction with coefficients $\alpha^{\prime}_{1}=1$ and $\alpha^{\prime}_{2n}=g_{n}$, $\alpha^{\prime}_{2n+1}=1-g_{n}$ for $n\geq 1$. Since all these coefficients are nonnegative, it follows that $\widetilde{{\bm{a}}}$ is a Stieltjes moment sequence. Therefore $\widetilde{\mu}$ is supported on $[0,3]$, so that $\widehat{\mu}=T_{-1}\widetilde{\mu}$ is supported on $[-1,2]$. But since $\widehat{\mu}$ is even, it must actually be supported on $[-1,1]$. Hence $\mu$ is supported on $[0,1]$, which shows that ${\bm{a}}$ is a Hausdorff moment sequence. $\square$

Acknowledgments

I wish to thank Alexander Dyachenko, Lily Liu, Mathias Pétréolle and Jiang Zeng for helpful conversations. This research was supported in part by Engineering and Physical Sciences Research Council grant EP/N025636/1.