6d. Stieltjes inversion

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We would like to end this chapter with an interesting application of the complex functions to probability theory. We have learned some basic probability in chapter 4, and in view of the material there, an interesting question is how to recover a probability measure out of its moments. And the answer here, which is non-trivial, is as follows:

Theorem

The density of a real probability measure [math]\mu[/math] can be recaptured from the sequence of moments [math]\{M_k\}_{k\geq0}[/math] via the Stieltjes inversion formula

[[math]] d\mu (x)=\lim_{t\searrow 0}-\frac{1}{\pi}\,Im\left(G(x+it)\right)\cdot dx [[/math]]

where the function on the right, given in terms of moments by

[[math]] G(\xi)=\xi^{-1}+M_1\xi^{-2}+M_2\xi^{-3}+\ldots [[/math]]

is the Cauchy transform of the measure [math]\mu[/math].

Show Proof

The Cauchy transform of our measure [math]\mu[/math] is given by:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty M_k\xi^{-k}\\\ &=&\int_\mathbb R\frac{\xi^{-1}}{1-\xi^{-1}y}\,d\mu(y)\\ &=&\int_\mathbb R\frac{1}{\xi-y}\,d\mu(y) \end{eqnarray*} [[/math]]

Now with [math]\xi=x+it[/math], we obtain the following formula:

[[math]] \begin{eqnarray*} Im(G(x+it)) &=&\int_\mathbb RIm\left(\frac{1}{x-y+it}\right)d\mu(y)\\ &=&\int_\mathbb R\frac{1}{2i}\left(\frac{1}{x-y+it}-\frac{1}{x-y-it}\right)d\mu(y)\\ &=&-\int_\mathbb R\frac{t}{(x-y)^2+t^2}\,d\mu(y) \end{eqnarray*} [[/math]]

By integrating over [math][a,b][/math] we obtain, with the change of variables [math]x=y+tz[/math]:

[[math]] \begin{eqnarray*} \int_a^bIm(G(x+it))dx &=&-\int_\mathbb R\int_a^b\frac{t}{(x-y)^2+t^2}\,dx\,d\mu(y)\\ &=&-\int_\mathbb R\int_{(a-y)/t}^{(b-y)/t}\frac{t}{(tz)^2+t^2}\,t\,dz\,d\mu(y)\\ &=&-\int_\mathbb R\int_{(a-y)/t}^{(b-y)/t}\frac{1}{1+z^2}\,dz\,d\mu(y)\\ &=&-\int_\mathbb R\left(\arctan\frac{b-y}{t}-\arctan\frac{a-y}{t}\right)d\mu(y) \end{eqnarray*} [[/math]]

Now observe that with [math]t\searrow0[/math] we have:

[[math]] \lim_{t\searrow0}\left(\arctan\frac{b-y}{t}-\arctan\frac{a-y}{t}\right) =\begin{cases} \frac{\pi}{2}-\frac{\pi}{2}=0& (y \lt a)\\ \frac{\pi}{2}-0=\frac{\pi}{2}& (y=a)\\ \frac{\pi}{2}-(-\frac{\pi}{2})=\pi& (a \lt y \lt b)\\ 0-(-\frac{\pi}{2})=\frac{\pi}{2}& (y=b)\\ -\frac{\pi}{2}-(-\frac{\pi}{2})=0& (y \gt b) \end{cases} [[/math]]

We therefore obtain the following formula:

[[math]] \lim_{t\searrow0}\int_a^bIm(G(x+it))dx=-\pi\left(\mu(a,b)+\frac{\mu(a)+\mu(b)}{2}\right) [[/math]]

Thus, we are led to the conclusion in the statement.

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Before getting further, let us mention that the above result does not fully solve the moment problem, because we still have the question of understanding when a sequence of numbers [math]M_1,M_2,M_3,\ldots[/math] can be the moments of a measure [math]\mu[/math]. We have here:

Theorem

A sequence of numbers [math]M_0,M_1,M_2,M_3,\ldots\in\mathbb R[/math], with [math]M_0=1[/math], is the series of moments of a real probability measure [math]\mu[/math] precisely when:

[[math]] \begin{vmatrix}M_0\end{vmatrix}\geq0\quad,\quad \begin{vmatrix} M_0&M_1\\ M_1&M_2 \end{vmatrix}\geq0\quad,\quad \begin{vmatrix} M_0&M_1&M_2\\ M_1&M_2&M_3\\ M_2&M_3&M_4\\ \end{vmatrix}\geq0\quad,\quad \ldots [[/math]]

That is, the associated Hankel determinants must be all positive.

Show Proof

This is something a bit more advanced, the idea being as follows:

(1) As a first observation, the positivity conditions in the statement tell us that the following associated linear forms must be positive:

[[math]] \sum_{i,j=1}^nc_i\bar{c}_jM_{i+j}\geq0 [[/math]]

(2) But this is something very classical, in one sense the result being elementary, coming from the following computation, which shows that we have positivity indeed:

[[math]] \int_\mathbb R\left|\sum_{i=1}^nc_ix^i\right|^2d\mu(x) =\int_\mathbb R\sum_{i,j=1}^nc_i\bar{c}_jx^{i+j}d\mu(x) =\sum_{i,j=1}^nc_i\bar{c}_jM_{i+j} [[/math]]

(3) As for the other sense, here the result comes once again from the above formula, this time via some standard functional analysis.

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Getting back now to more concrete things, the point is that we have: \begin{fact} Given a graph [math]X[/math], with distinguished vertex [math]*[/math], we can talk about the probability measure [math]\mu[/math] having as [math]k[/math]-th moment the number of length [math]k[/math] loops based at [math]*[/math]:

[[math]] M_k=\Big\{\!*-i_1-i_2-\ldots-i_k=*\Big\} [[/math]]

As basic examples, for the graph [math]\mathbb N[/math] the moments must be the Catalan numbers [math]C_k[/math], and for the graph [math]\mathbb Z[/math], the moments must be the central binomial coefficients [math]D_k[/math]. \end{fact} To be more precise, the first assertion, regarding the existence and uniqueness of [math]\mu[/math], follows from a basic linear algebra computation, by diagonalizing the adjacency matrix of [math]X[/math]. As for the examples, for the graph [math]\mathbb N[/math] we more or less already know this, from our various Catalan number considerations from chapter 2, and for the graph [math]\mathbb Z[/math] this is something elementary, that we will leave here as an instructive exercise.

Needless to say, counting loops on graphs, as in Fact 6.34, is something important in applied mathematics, and physics. So, back to our business now, motivated by all this, as a basic application of the Stieltjes formula, let us solve the moment problem for the Catalan numbers [math]C_k[/math], and for the central binomial coefficients [math]D_k[/math]. We first have:

Theorem

The real measure having as even moments the Catalan numbers, [math]C_k=\frac{1}{k+1}\binom{2k}{k}[/math], and having all odd moments [math]0[/math] is the measure

[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

called Wigner semicircle law on [math][-2,2][/math].

Show Proof

In order to apply the inversion formula, our starting point will be the formula from chapter 3 for the generating series of the Catalan numbers, namely:

[[math]] \sum_{k=0}^\infty C_kz^k=\frac{1-\sqrt{1-4z}}{2z} [[/math]]

By using this formula with [math]z=\xi^{-2}[/math], we obtain the following formula:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty C_k\xi^{-2k}\\ &=&\xi^{-1}\cdot\frac{1-\sqrt{1-4\xi^{-2}}}{2\xi^{-2}}\\ &=&\frac{\xi}{2}\left(1-\sqrt{1-4\xi^{-2}}\right)\\ &=&\frac{\xi}{2}-\frac{1}{2}\sqrt{\xi^2-4} \end{eqnarray*} [[/math]]

Now let us apply Theorem 6.32. The study here goes as follows:

(1) According to the general philosophy of the Stieltjes formula, the first term, namely [math]\xi/2[/math], which is “trivial”, will not contribute to the density.

(2) As for the second term, which is something non-trivial, this will contribute to the density, the rule here being that the square root [math]\sqrt{\xi^2-4}[/math] will be replaced by the “dual” square root [math]\sqrt{4-x^2}\,dx[/math], and that we have to multiply everything by [math]-1/\pi[/math].

(3) As a conclusion, by Stieltjes inversion we obtain the following density:

[[math]] d\mu(x) =-\frac{1}{\pi}\cdot-\frac{1}{2}\sqrt{4-x^2}\,dx =\frac{1}{2\pi}\sqrt{4-x^2}dx [[/math]]

Thus, we have obtained the mesure in the statement, and we are done.

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We have the following version of the above result:

Theorem

The real measure having as sequence of moments the Catalan numbers, [math]C_k=\frac{1}{k+1}\binom{2k}{k}[/math], is the measure

[[math]] \pi_1=\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx [[/math]]

called Marchenko-Pastur law on [math][0,4][/math].

Show Proof

As before, we use the standard formula for the generating series of the Catalan numbers. With [math]z=\xi^{-1}[/math] in that formula, we obtain the following formula:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty C_k\xi^{-k}\\ &=&\xi^{-1}\cdot\frac{1-\sqrt{1-4\xi^{-1}}}{2\xi^{-1}}\\ &=&\frac{1}{2}\left(1-\sqrt{1-4\xi^{-1}}\right)\\ &=&\frac{1}{2}-\frac{1}{2}\sqrt{1-4\xi^{-1}} \end{eqnarray*} [[/math]]

With this in hand, let us apply now the Stieltjes inversion formula, from Theorem 6.32. We obtain, a bit as before in Theorem 6.35, the following density:

[[math]] d\mu(x) =-\frac{1}{\pi}\cdot-\frac{1}{2}\sqrt{4x^{-1}-1}\,dx =\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx [[/math]]

Thus, we are led to the conclusion in the statement.

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Regarding now the central binomial coefficients, we have here:

Theorem

The real probability measure having as moments the central binomial coefficients, [math]D_k=\binom{2k}{k}[/math], is the measure

[[math]] \alpha_1=\frac{1}{\pi\sqrt{x(4-x)}}\,dx [[/math]]

called arcsine law on [math][0,4][/math].

Show Proof

We have the following computation, using formulae from chapter 3:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty D_k\xi^{-k}\\ &=&\frac{1}{\xi}\sum_{k=0}^\infty D_k\left(-\frac{t}{4}\right)^k\\ &=&\frac{1}{\xi}\cdot\frac{1}{\sqrt{1-4/\xi}}\\ &=&\frac{1}{\sqrt{\xi(\xi-4)}} \end{eqnarray*} [[/math]]

But this gives the density in the statement, via Theorem 6.32.

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Finally, we have the following version of the above result:

Theorem

The real probability measure having as moments the middle binomial coefficients, [math]E_k=\binom{k}{[k/2]}[/math], is the following law on [math][-2,2][/math],

[[math]] \sigma_1=\frac{1}{2\pi}\sqrt{\frac{2+x}{2-x}}\,dx [[/math]]

called modified the arcsine law on [math][-2,2][/math].

Show Proof

In terms of the central binomial coefficients [math]D_k[/math], we have:

[[math]] E_{2k}=\binom{2k}{k}=\frac{(2k)!}{k!k!}=D_k [[/math]]

[[math]] E_{2k-1}=\binom{2k-1}{k}=\frac{(2k-1)!}{k!(k-1)!}=\frac{D_k}{2} [[/math]]

Standard calculus based on the Taylor formula for [math](1+t)^{-1/2}[/math] gives:

[[math]] \frac{1}{2x}\left(\sqrt{\frac{1+2x}{1-2x}}-1\right)=\sum_{k=0}^\infty E_kx^k [[/math]]

With [math]x=\xi^{-1}[/math] we obtain the following formula for the Cauchy transform:

[[math]] \begin{eqnarray*} G(\xi) &=&\xi^{-1}\sum_{k=0}^\infty E_k\xi^{-k}\\ &=&\frac{1}{\xi}\left(\sqrt{\frac{1+2/\xi}{1-2/\xi}}-1\right)\\ &=&\frac{1}{\xi}\left(\sqrt{\frac{\xi+2}{\xi-2}}-1\right) \end{eqnarray*} [[/math]]

By Stieltjes inversion we obtain the density in the statement.

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All this is very nice, and we are obviously building here, as this book goes by, some good knowledge in probability theory. We will be back to all this later.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].