{"id":22252,"date":"2025-11-22T20:48:52","date_gmt":"2025-11-22T19:48:52","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=22252"},"modified":"2026-02-12T16:01:46","modified_gmt":"2026-02-12T15:01:46","slug":"back-to-basics-frechet-shohat-cumulants-clt","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2025\/11\/22\/back-to-basics-frechet-shohat-cumulants-clt\/","title":{"rendered":"Back to basics : Fr\u00e9chet-Shohat cumulants CLT"},"content":{"rendered":"
\"\"<\/a>
Thorvald Nicolai Thiele (1838 - 1910) who is credited for the first introduction of the cumulants, in 1889, under the name of semi-invariants. The name cumulants seems to be due to Ronald Fisher, John Wishart, or Harold Hotelling.<\/figcaption><\/figure>\n

This post is about a Central Limit Theorem (CLT) involving cumulants, discovered in the 1930s by Ren\u00e9 Maurice Fr\u00e9chet and James Alexander Shohat. It is simple and useful.<\/p>\n

Cumulants.<\/strong> If $X$ is a real random variable with finite moments of all orders, its cumulants ${(C_k(X))}_{k\\geq1}$ are given by the derivatives of the logarithm of its characteristic function, namely \\[ C_k(X)=(-\\mathrm{i})^k\\partial^k_{t=0}\\log\\mathbb{E}(\\mathrm{e}^{\\mathrm{i}tX}). \\] The continuous branch of the logarithm used here is the one such that $\\log(1)=0$. If the characteristic function is analytic in a neighborhood $O$ of the origin, then for all $t\\in O$, \\[ \\log\\mathbb{E}(\\mathrm{e}^{\\mathrm{i}tX})=\\sum_{k=1}^\\infty\\frac{\\mathrm{i}^kt^k}{k!}C_k(X). \\] In particular \\[ C_1(X)=\\mathbb{E}(X) \\quad\\text{and}\\quad C_2(X)=\\mathrm{Var}(X)=\\mathbb{E}(X^2)-\\mathbb{E}(X)^2. \\] More generally, for all $k$, there exists a universal polynomial that depends only on $k$ expressing $C_k(X)$ in terms of the moments of $X$ up to order $k$, and conversely. This universal algebraic link can be used as a definition of the cumulants from the moments. If the Laplace transform of $X$ exists in a neighborhood $O$ of the origin (finite $\\pm$ exponential moments) then for all $t\\in O$, \\[ \\log\\mathbb{E}(\\mathrm{e}^{tX}) =\\sum_{k=1}^\\infty\\frac{(-1)^kt^k}{k!}C_k(X) \\quad\\text{and}\\quad C_k(X)=(-1)^k\\partial^k_{t=0}\\log\\mathbb{E}(\\mathrm{e}^{tX}). \\] The cumulants can be seen as more attractive than the moments due to their translation-dilation properties and additivity property with respect to independence, namely \\[ C_k(\\lambda X)=\\lambda^kC_k(X),\\quad C_k(X+c)=C_k(X)+c\\mathbb{1}_{k=1} \\] and \\[ C_k(X+Y)=C_k(X)+C_k(Y) \\quad\\text{if $X$ and $Y$ are independent.} \\] We have $X\\sim\\mathcal{N}(0,1)$ if and only if $C_1(X)=0$, $C_2(X)=1$, and $C_k(X)=0$ for all $k\\geq3$.<\/p>\n

The Fr\u00e9chet-Shohat theorem : a CLT from the cumulants.<\/strong> Let ${(X_n)}_{n\\geq1}$ be a sequence of real random variables with finite moments of all orders, and such that \\[ \\lim_{n\\to\\infty}\\frac{C_k(X_n)}{\\sqrt{C_2(X_n)}^k}=0 \\quad\\text{for all $k\\geq3$}. \\] Then the following CLT holds: \\[ \\frac{X_n-\\mathbb{E}(X_n)}{\\sqrt{\\mathrm{Var}(X_n)}} \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathcal{N}(0,1). \\]<\/p>\n

About the condition.<\/strong> It is satisfied in the usual iid case $X_n=Z_1+\\cdots+Z_n$ with $Z_1,\\ldots,Z_n$ iid with positive variance and finite moments of all orders. Indeed, $C_k(X_n)=nC_k(Z_1)$, hence \\[ C_2(X_n)=\\mathrm{Var}(Z_1)n\\xrightarrow[n\\to\\infty]{}+\\infty \\] while for $k\\geq3$, \\[ \\frac{C_k(X_n)}{\\sqrt{C_2(X_n)}^k} =\\frac{C_k(Z_1)}{\\sqrt{C_2(Z_1)}^k}n^{1-\\frac{k}{2}} \\xrightarrow[n\\to\\infty]{} 0. \\] Note that the condition on the cumulants is also satisfied when the variance blows up while the higher order cumulants remain bounded, as $n\\to\\infty$, more precisely \\[ \\lim_{n\\to\\infty}C_2(X_n)=+\\infty \\quad\\text{while}\\quad \\sup_{n\\geq1}C_k(X_n) < \\infty \\quad\\text{for all $k\\geq3$}, \\] a situation that can be encountered for certain linear statistics of point processes.<\/p>\n

Usefulness.<\/strong> For certain stochastic models, it is possible to express the cumulants using combinatorics, paving the way to the Fr\u00e9chet-Shohat theorem. This is for instance the case for determinantal point processes, as observed initially by Costin and Lebowitz, and further studied notably by Soshnikov, and by Rider and Vir\u00e1g.<\/p>\n

Proof of the theorem.<\/strong> If we define \\[ Y_n=\\frac{X_n-\\mathbb{E}(X_n)}{\\sqrt{\\mathrm{Var}(X_n)}}=\\frac{X_n-C_1(X_n)}{\\sqrt{C_2(X_n)}}, \\] then we get, by the basic translation-scaling properties of cumulants \\[ C_k(Y_n) =\\begin{cases} 0 & \\text{if $k=1$} \\\\ 1 & \\text{if $k=2$} \\\\ \\displaystyle\\frac{C_k(X_n)}{\\sqrt{C_2(X_n)}^k} & \\text{if $k\\geq3$} \\end{cases}. \\] Thus $Y_n\\to\\mathcal{N}(0,1)$ as $n\\to\\infty$, in the sense of cumulants, and therefore also in the sense of moments. Now the standard normal $\\mathcal{N}(0,1)$ is characterized by its moments, according to the Carleman condition, and therefore, the convergence in the sense of moments implies the convergence in distribution, see below.<\/p>\n

The Carleman condition on characterization by the moments.<\/strong> Let $\\mu$ be a probability measure on $\\mathbb{R}$ with finite moments of all orders : $\\mathbb{R}[X]\\subset L^1(\\mu)$. We denote them \\[ m_k=\\int x^k\\mathrm{d}\\mu(x),\\quad k\\geq1. \\] If (Carleman condition) \\[ \\sum_k m_{2k}^{-\\frac{1}{2k}}=\\infty, \\] then $\\mu$ is the unique probability measure on $\\mathbb{R}$ with moments ${(m_k)}_{k\\geq1}$, in other words, it is characterized by its moments.<\/p>\n

We say then that the Hamburger moment problem, which is the moment problem on the real line, is determinate. Actually the Carleman condition is essentially a condition from complex analysis that implies the quasi-analycity of the Fourier transform. See for instance [Feller, p. 222].<\/p>\n

The Carleman condition is obviously satisfied if $\\mu$ has compact support since in this case \\[ m_{2k}\\leq C^{2k}. \\] Beyond compact support, the Carleman condition is satisfied by say $\\mathcal{N}(0,1)$ since in this case \\[ m_{2k}=(2k-1)!!=\\frac{(2k)!}{2^kk!}\\sim_{k\\to\\infty}\\sqrt{2}(\\frac{2k}{\\mathrm{e}})^k. \\] However the log-normal distribution, which is the law of $\\mathrm{e}^Z$ when $Z\\sim\\mathcal{N}(0,1)$, has a so heavy tail that it does not satisfy the Carleman condition. Indeed, in this case \\[ m_{2k}=\\mathrm{e}^{\\frac{k^2}{2}}. \\] It can also be shown that it is not characterized by its moments. Actually a probability measure may fail to satisfy the Carleman condition while being caracterized by its moments.<\/p>\n

From convergence of moments to narrow convergence.<\/strong> Let ${(\\mu_n)}_n$ and $\\mu$ be probability measures on $\\mathbb{R}$ with finite moments of all orders, denoted $m_{n,k}$ and $m_k$ respectively. If $\\mu$ is characterized by its moments (for instance it satisfies the Carleman condition!) and if \\[ \\lim_{n\\to\\infty}m_{n,k}=m_k\\quad\\text{for all $k$} \\] then \\[ \\mu_n\\to\\mu\\quad\\text{narrowly as}\\quad n\\to\\infty, \\] in other words weakly with respect to continuous and bounded test functions.<\/p>\n

Indeed, since the sequence of second moments ${(m_{n,2})_n}$ is bounded, by the Markov inequality, the sequence ${(\\mu_n)}_n$ is tight. Therefore, by the Prohorov theorem, it is sequentially relatively compact for the narrow convergence. Thus it suffices to show that all converging sub-sequences have the same limit. Now, suppose that a sub-sequence ${(\\mu_{\\varphi(n)})}_n$ converges towards a probability measure $\\nu$. For all fixed $k$, since ${(m_{n,2k})}_n$ is bounded, it follows that for all $k'< 2k$, $\\nu$ admits a finite moment of order $k'$, denoted $M_{k'}$, and $\\lim_{n\\to\\infty}m_{n,k'}=M_{k'}$. But $M_{k'}=m_{k'}$, and thus $\\nu=\\mu$ since $\\mu$ is characterized by its moments. This is what is written in [Billingsley, Th. 30.2] for instance.<\/p>\n

Convergence in the sense of moments and Wasserstein distances.<\/strong> Let $\\mathcal{P}_k$ be the set of probability measures with finite moments up to order $k$. Recall that the Wasserstein distance of order $k$ on $\\mathcal{P}_k$ is defined for all $\\mu,\\nu\\in\\mathcal{P}_k$ by \\[ \\mathrm{W}_k(\\mu,\\nu)=\\inf\\mathbb{E}(|X-Y|^k)^{1\/k} \\] where the infimum runs over all couples of random variables $(X,Y)$ with $X\\sim\\mu$ and $Y\\sim\\nu$. It turns out that for all $k$, if ${(\\mu_n)}_n$ and $\\mu$ are in $\\mathcal{P}_k$ then \\[ \\mathrm{W}_k(\\mu_n,\\mu)\\xrightarrow[n\\to\\infty]{}0 \\] if and only if $\\mu_n\\to\\mu$ narrowly as well as in the sense of moments up to order $k$.<\/p>\n

It follows that if ${(\\mu_n)}_n$ and $\\mu$ are probability measures on $\\mathbb{R}$ with finite moments of all orders, in other words in $\\cap_k\\mathcal{P}_k$, and if $\\mu$ is characterized by its moments, then \\[ \\mu_n\\to\\mu\\quad\\text{in the sense of moments} \\] if and only if \\[ \\mathrm{W}_k(\\mu_n,\\mu)\\xrightarrow[n\\to\\infty]{}0\\quad\\text{for all $k$.} \\]<\/p>\n

Further reading.<\/strong><\/p>\n

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