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De la Vallée Poussin on Uniform Integrability

Charles Jean de la Vallée-Poussin
Charles de la Vallée-Poussin.

This post is devoted to some probabilistic aspects of uniform integrability, a basic concept that I like very much. Let \( {\Phi} \) be the class of non-decreasing functions \( {\varphi:\mathbb{R}_+\rightarrow\mathbb{R}_+} \) such that

\[ \lim_{x\rightarrow+\infty}\frac{\varphi(x)}{x}=+\infty. \]

This class contains for instance the convex functions \( {x\mapsto x^p} \) with \( {p>1} \) and \( {x\mapsto x\log(x)} \). Let us fix a probability space \( {(\Omega,\mathcal{A},\mathbb{P})} \), and denote by \( {L^\varphi} \) the set of random variables \( {X} \) such that \( {\varphi(|X|)\in L^1=L^1((\Omega,\mathcal{F},\mathbb{P}),\mathbb{R})} \). We have \( {L^\varphi\subsetneq L^1} \). Clearly, if \( {{(X_i)}_{i\in I}\subset L^1} \) is bounded in \( {L^\varphi} \) with \( {\varphi\in\Phi} \) then \( {{(X_i)}_{i\in I}} \) is bounded in \( {L^1} \).

Uniform integrability. For any family \( {{(X_i)}_{i\in I}\subsetneq L^1} \), the following three properties are equivalent. When one (and thus all) of these properties holds true, we say that the family \( {{(X_i)}_{i\in I}} \) is uniformly integrable (UI). The first property can be seen as a natural definition of uniform integrability.

  1. (definition of UI) \( {\lim_{x\rightarrow+\infty}\sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x})=0} \);
  2. (epsilon-delta) the family is bounded in \( {L^1} \): \( {\sup_{i\in I}\mathbb{E}(|X_i|)<\infty} \), and moreover \( {\forall\varepsilon>0} \), \( {\exists\delta>0} \), \( {\forall A\in\mathcal{F}} \), \( {\mathbb{P}(A)\leq\delta\Rightarrow\sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_A)\leq\varepsilon} \);
  3. (de la Vallée Poussin) there exists \( {\varphi\in\Phi} \) such that \( {\sup_{i\in I}\mathbb{E}(\varphi(|X_i|))<\infty} \).

The second property is often referred to as the “epsilon-delta” criterion. The third and last property is a boundedness in \( {L^\varphi\subsetneq L^{1}} \) and is due to Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin (1866 – 1962), a famous Belgian mathematician, also well known for his proof of the prime number theorem.

Proof of \( {1\Rightarrow2} \). For the boundedness in \( {L^1} \), we write, for any \( {i\in I} \), and some \( {x\geq0} \) large enough,

\[ \mathbb{E}(|X_i|) \leq \mathbb{E}(|X_i|\mathbf{1}_{|X_i|<x})+\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x}) \leq x+\sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x})<\infty. \]

Next, by assumption, for any \( {\varepsilon>0} \), there exists \( {x_\varepsilon} \) such that \( {\sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x})\leq\varepsilon} \) for any \( {x\geq x_\varepsilon} \). If \( {\mathbb{P}(A)\leq \delta_\varepsilon:=\varepsilon/x_\varepsilon} \) then for any \( {i\in I} \),

\[ \mathbb{E}(|X_i|\mathbf{1}_A) =\mathbb{E}(|X_i|\mathbf{1}_{|X_i|<x_\varepsilon}\mathbf{1}_A) +\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x_\varepsilon}\mathbf{1}_A) \leq x_\varepsilon\mathbb{P}(A)+\varepsilon\leq 2\varepsilon. \]

Proof of \( {2\Rightarrow1} \). Since \( {{(X_i)}_{i\in I}} \) is bounded in \( {L^1} \), for every \( {j\in I} \), if \( {A:=A_j:=\{|X_j|\geq x\}} \), then, by Markov inequality, \( {\mathbb{P}(A)\leq x^{-1}\sup_{i\in I}\mathbb{E}(|X_i|)\leq \delta} \) for \( {x} \) large enough, uniformly in \( {j\in I} \), and the assumption gives \( {\lim_{x\rightarrow+\infty}\sup_{i,j\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_j|\geq x})=0} \).

Proof of \( {3\Rightarrow1} \). For any \( {\varepsilon>0} \), since \( {\varphi\in\Phi} \), by definition of \( {\Phi} \) there exists \( {x_\varepsilon\geq0} \) such that \( {x\leq\varepsilon \varphi(x)} \) for every \( {x\geq x_\varepsilon} \), and therefore

\[ \sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x_\varepsilon}) \leq \varepsilon\sup_{i\in I}\mathbb{E}(\varphi(|X_i|)\mathbf{1}_{|X_i|\geq x_\varepsilon}) \leq \varepsilon\sup_{i\in I}\mathbb{E}(\varphi(|X_i|)). \]

Proof of \( {1\Rightarrow 3} \). Let us seek for a piecewise linear function \( {\varphi} \) of the form

\[ \varphi(x)=\int_0^x\varphi'(t)dt \quad\mbox{with}\quad \varphi’=\sum_{n=1}^\infty u_n\mathbf{1}_{[n,n+1[} \quad\mbox{and}\quad u_n=\sum_{m\geq1}\mathbf{1}_{x_m\leq n} \]

for a sequence \( {{(x_m)}_{m\geq1}\nearrow+\infty} \) to be constructed. We have \( {\varphi\in\Phi} \) since \( {\lim_{n\rightarrow\infty}u_n=+\infty} \). Moreover, since \( {\varphi\leq\sum_{n=1}^\infty(u_1+\cdots+u_n)\mathbf{1}_{[n,n+1[}} \), we get, for any \( {i\in I} \),

\[ \begin{array}{rcl} \mathbb{E}(\varphi(|X_i|)) &\leq& \sum_{n=1}^\infty (u_1+\cdots+u_n)\mathbb{P}(n\leq |X_i|<n+1) \\ &=& \sum_{n=1}^\infty u_n\mathbb{P}(|X_i|\geq n)\\ &=& \sum_{m\geq1}\sum_{n\geq x_m}\mathbb{P}(|X_i|\geq n)\\ &\leq & \sum_{m\geq1}\sum_{n\geq x_m}n\mathbb{P}(n\leq |X_i|<n+1)\\ &\leq & \sum_{m\geq1}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x_m})\\ &\overset{*}{\leq} & \sum_{m\geq1}2^{-m}<\infty \end{array} \]

where \( {\overset{*}{\leq}} \) holds if for every \( {m} \) we select \( {x_m} \) such that \( {\sup_{i\in I}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x_m})\leq 2^{-m}} \), which is allowed by assumption (we may replace \( {2^{-m}} \) by anything sumable in \( {m} \)).

This achieves the proof of the equivalence of the three properties: \( {1\Leftrightarrow 2\Leftrightarrow 3} \).

Alternative proof of \( {1\Rightarrow 3} \). Following a suggestion made by Nicolas Fournier on an earlier version of this post, one can simply construct \( {\varphi} \) as follows:

\[ \varphi(x)=\sum_{m\geq 1} (x-x_m)_+. \]

This function is non-decreasing and convex, and

\[ \frac{\varphi(x)}{x} = \sum_{m\geq1} \left(1-\frac{x_m}{x}\right)_+ \underset{x\rightarrow\infty}{\nearrow} \sum_{m\geq1} 1=\infty. \]

It remains to note that

\[ \mathbb{E}(\varphi(|X_i|)) \leq \sum_{m\geq1}\mathbb{E}(|X_i|\mathbf{1}_{|X_i|\geq x_m}) \leq 1. \]

Convexity and moderate growth. In the de la Vallée Poussin criterion, one can always assume that the function \( {\varphi} \) is convex (in particular continuous), with moderate growth in the sense that for every \( {x\geq0} \),

\[ \varphi(x)\leq x^2. \]

Indeed, the construction of \( {\varphi} \) that we gave above provides a function \( {\varphi} \) with piecewise constant and non-decreasing derivative, thus the function is convex (it is a Young function). Following Paul-André Meyer, to derive the moderate growth property, we may first observe that thanks to the way we constructed \( {\varphi} \), for every \( {x\geq0} \),

\[ \varphi'(2x)\leq c\varphi'(x) \]

where one can take \( {c=2} \). This follows from the fact that we have the freedom to take the \( {x_m} \)’s as large as we want, for instance in such a way that for every \( {n\geq1} \),

\[ u_{2n}\leq u_n+\sum_{m\geq1}\mathbf{1}_{n<x_m\leq 2n}\leq 2u_n. \]

Consequently, the function \( {\varphi} \) itself has also moderate growth since, denoting \( {C:=2c} \),

\[ \varphi(2x)=\int_0^{2x}\!g(u)\,du=\int_0^x\!g(2t)\,2dt\leq 2c\varphi(x)=C\varphi(x). \]

Now \( {\varphi(x)\leq C^{1+k}\varphi(2^{-k}x)} \) for any \( {k\geq1} \), and taking \( {k=k_x=\lceil\log_2(x)\rceil} \) we obtain

\[ \varphi(x)\leq C^2C^{\log_2(x)}\varphi(1)=C^2\varphi(1)2^{\log_2(C)\log_2(x)}=C^2\varphi(1)x^{\log_2(C)}. \]

Since one can take \( {c=2} \) we get \( {C=4} \), which allows \( {\varphi(x)\leq x^2} \) by scaling.

Examples of UI families.

  • Any finite subset of \( {L^1} \) is UI;
  • More generally, if \( {\sup_{i\in I}|X_i|\in L^1} \) (domination: \( {|X_i|\leq X} \) for every \( {i\in I} \), with \( {X\in L^1} \)) then \( {{(X_i)}_{i\in I}} \) is UI. To see it, we may first observe that the singleton family \( {\{\sup_{i\in I}|X_i|\}} \) is UI, and thus, by the de la Vallée Poussin criterion, there exists \( {\varphi\in\Phi} \) such that \( {\varphi(\sup_{i\in I}|X_i|)\in L^1} \), and therefore

    \[ \sup_{i\in I}\mathbb{E}(\varphi(|X_i|))\leq\mathbb{E}(\sup_{i\in I}\varphi(|X_i|))\leq\mathbb{E}(\varphi(\sup_{i\in I}|X_i|))<\infty, \]

    which implies, by the de la Vallée Poussin criterion again, that \( {{(X_i)}_{i\in I}} \) is UI;

  • If \( {\mathcal{U}_1\subset L^1,\ldots,\mathcal{U}_n\subset L^1} \) is a finite collection of UI families then the union \( {\mathcal{U}_1\cup\cdots\cup\mathcal{U}_n} \) is UI.
  • If \( {\mathcal{U}} \) is UI then its convex envelope (or convex hull) is UI. Beware however that the vector span of \( {\mathcal{U}} \) is not UI in general.
  • If \( {{(X_n)}_{n\geq1},X\in L^1} \) and \( {X_n\overset{L^1}{\rightarrow} X} \) then \( {{(X_n)}_{n\geq1}\cup\{X\}} \) is UI and \( {{(X_n-X)}_{n\geq1}} \) is UI. To see it, for any \( {\varepsilon>0} \), we first select \( {n} \) large enough such that \( {\sup_{k\geq n}\mathbb{E}(|X_k-X|)\leq\varepsilon} \), and then \( {\delta>0} \) with the epsilon-delta criterion for the finite family \( {\{X_1,\ldots,X_n\}} \), which gives, for any \( {A\in\mathcal{F}} \) such that \( {\mathbb{P}(A)\leq\delta} \),

    \[ \sup_{n\geq1}\mathbb{E}(|X_n-X|\mathbf{1}_A) \leq \max\left(\max_{1\leq k\leq n}\mathbb{E}(|X_k-X|\mathbf{1}_A) ; \sup_{k\geq n}\mathbb{E}(|X_k-X|) \right) \leq \varepsilon. \]

  • The de la Vallée Poussin criterion is often used with \( {\varphi(x)=x^2} \), and means in this case that every bounded subset of \( {L^2} \) is UI.

Integrability. The de la Vallée Poussin criterion, when used with a singleton UI family \( {\{X\}} \), states that \( {X\in L^1} \) implies that \( {\varphi(|X|)\in L^1} \) for some \( {\varphi\in\Phi} \). In other words, for every random variable, integrability can always be improved in a sense. There is not any paradox here since \( {\varphi} \) depends actually on \( {X} \). Topologically, integrability is in a sense an open statement rather than a closed statement. An elementary instance of this phenomenon is visible for Riemann series, in the sense that if \( {\sum_{n\geq1}n^{-s}<\infty} \) for some \( {s>0} \) then \( {\sum_{n\geq1}n^{-s’}<\infty} \) for some \( {s'<s} \), because the convergence condition of the series is “\( {s>1} \)”, which is an open condition.

Integrability of the limit. If \( {{(X_n)}_{n\geq1}} \) is UI and \( {X_n\rightarrow X} \) almost surely as \( {n\rightarrow\infty} \) then \( {X\in L^1} \). Indeed, by the Fatou Lemma,

\[ \mathbb{E}(|X|) =\mathbb{E}(\varliminf_{n\rightarrow\infty}|X_n|) \leq\varliminf_{n\rightarrow\infty}\mathbb{E}(|X_n|) \leq\sup_{n\geq1}\mathbb{E}(|X_n|)<\infty. \]

Dominated convergence. For any \( {{(X_n)}_{n\geq1},X\in L^1} \), we have

\[ X_n\overset{ L^1}{\rightarrow}X \quad\text{if and only if}\quad X_n\overset{\mathbb{P}}{\rightarrow}X \text{ and } {(X_n)}_{n\geq1}\text{ is UI} \]

and in fact, for any \( {{(X_n)}_{n\geq1}\in L^1} \) and any random variable \( {X} \),

\[ X\in L^1\text{ and }X_n\overset{ L^1}{\rightarrow}X \quad\text{if and only if}\quad X_n\overset{\mathbb{P}}{\rightarrow}X \text{ and } {(X_n)}_{n\geq1}\text{ is UI}. \]

This can be seen as an improved dominated convergence theorem (since \( {{(X_n)}_{n\geq1}} \) is UI when \( {\sup_n|X_n|\in L^1} \)). The proof may go as follows. We already know that if \( {X_n\rightarrow X} \) in \( {L^1} \) then \( {X_n\rightarrow X} \) in probability (Markov inequality) and \( {{(X_n)}_{n\geq1}\cup\{X\}} \) is UI (see above). Conversely, if \( {X_n\rightarrow X} \) in probability and \( {{(X_n)}_{n\geq1}} \) is UI, then \( {{(X_n-X)}_{n\geq1}} \) is UI since \( {X\in L^1} \), and thus, using the convergence in probability and the epsilon-delta criterion, we obtain that for any \( {\varepsilon>0} \) and large enough \( {n} \),

\[ \mathbb{E}(|X_n-X|) =\mathbb{E}(|X_n-X|\mathbf{1}_{|X_n-X|\geq\varepsilon}) +\mathbb{E}(|X_n-X|\mathbf{1}_{|X_n-X|<\varepsilon}) \leq 2\varepsilon. \]

Martingales. The American mathematician Joseph Leo Doob (1910 – 2004) has shown that if a sub-martingale \( {{(M_n)}_{n\geq1}} \) is bounded in \( {L^1} \) then there exists \( {M_\infty\in L^1} \) such that \( {M_n\rightarrow M_\infty} \) almost surely. Moreover, in the case of martingales, this convergence holds also in \( {L^1} \) if and only if \( {{(M_n)}_{n\geq1}} \) is UI. In the same spirit, and a bit more precisely, if \( {{(M_n)}_{n\geq1}} \) is a martingale for a filtration \( {{(\mathcal{F}_n)}_{n\geq1}} \), then the following two properties are equivalent:

  • \( {{(M_n)}_{n\geq1}} \) is UI;
  • \( {{(M_n)}_{n\geq1}} \) is closed, meaning that there exists \( {M_\infty\in L^1} \) such that \( {M_n=\mathbb{E}(M_\infty|\mathcal{F}_n)} \) for all \( {n\geq1} \), and moreover \( {M_n\rightarrow M_\infty} \) almost surely and in \( {L^1} \).

Are there non UI martingales? Yes, but they are necessarily unbounded: \( {\sup_{n\geq0}|M_n|\not\in L^1} \), otherwise we may apply dominated convergence. A nice counter example is given by critical Galton-Watson branching process, defined recursively by \( {M_0=1} \) and

\[ M_{n+1}=X_{n+1,1}+\cdots+X_{n+1,M_n}, \]

where \( {X_{n+1,j}} \) is the number of offspring of individual \( {j} \) in generation \( {n} \), and \( {{(X_{j,k})}_{j,k\geq1}} \) are i.i.d. random variables on \( {\mathbb{N}} \) with law \( {\mu} \) of mean \( {1} \) and such that \( {\mu(0)>0} \). The sequence \( {{(M_n)}_{n\geq1}} \) is a non-negative martingale, and thus it converges almost surely to some \( {M_\infty\in L^1} \). It is also a Markov chain with state space \( {\mathbb{N}} \). The state \( {0} \) is absorbing and all the remaining states can lead to \( {0} \) and are thus transient. It follows then that almost surely, either \( {{(M_n)}_{n\geq0}} \) converges to \( {0} \) or to \( {+\infty} \), and since \( {M_\infty \in L^1} \), it follows that \( {M_\infty=0} \). However, the convergence cannot hold in \( {L^1} \) since this leads to the contradiction \( {1=\mathbb{E}(M_n)\rightarrow\mathbb{E}(M_\infty)=0} \) (note that \( {{(M_n)}_{n\geq1}} \) is bounded in \( {L^1} \)).

Topology. The Dunford-Pettis theorem, due to the American mathematicians Nelson James Dunford (1906 – 1986) and Billy James Pettis (1913 – 1979), states that for every family \( {{(X_i)}_{i\in I}\in L^1} \), the following propositions are equivalent.

  • \( {{(X_i)}_{i\in I}} \) is UI;
  • \( {{(X_i)}_{i\in I}} \) is relatively compact for the weak \( {\sigma(L^1,L^\infty)} \) topology;
  • \( {{(X_i)}_{i\in I}} \) is relatively sequentially compact for the weak \( {\sigma(L^1,L^\infty)} \) topology.

The proof, which is not given in this post, can be found for instance in Delacherie and Meyer or in Diestel. We just recall that a sequence \( {{(X_n)}_{n\geq1}} \) in \( {L^1} \) converges to \( {X\in L^1} \) for the weak \( {\sigma(L^1,L^\infty)} \) topology when \( {\lim_{n\rightarrow\infty}\ell(X_n)=\ell(X)} \) for every \( {\ell\in (L^1)’=L^\infty} \), in other words when \( {\lim_{n\rightarrow\infty}\mathbb{E}(YX_n)=\mathbb{E}(YX)} \) for every \( {Y\in L^\infty} \).

The Dunford-Pettis theorem opens the door for the fine analysis of closed (possibly linear) subsets of \( {L^1} \), a deep subject in functional analysis and Banach spaces.

Tightness. If \( {{(X_i)}_{i\in I}\subset L^1} \) is bounded in \( {L^\varphi} \) for \( {\varphi:\mathbb{R}_+\rightarrow\mathbb{R}_+} \) non-decreasing and such that \( {\lim_{x\rightarrow\infty}\varphi(x)=+\infty} \), then the Markov inequality gives

\[ \sup_{i\in I}\mathbb{P}(|X_i|\geq R)\leq \frac{\sup_{i\in I}\mathbb{E}(\varphi(|X_i|))}{\varphi(R)}\underset{R\rightarrow\infty}{\longrightarrow0} \]

and thus the family of distributions \( {{(\mathbb{P}_{X_i})}_{i \in I}} \) is tight, in the sense that for every \( {\varepsilon>0} \), there exists a compact subset \( {K_\varepsilon} \) of \( {\mathbb{R}} \) such that \( {\sup_{i\in I}\mathbb{P}(|X_i|\in K_\varepsilon)\geq 1-\varepsilon} \). The Prohorov theorem states that tightness is equivalent to being relatively compact for the narrow topology (which is by the way metrizable by the bounded-Lipschitz Fortet-Mourier distance). The Prohorov and the Dunford-Pettis theorems correspond to different topologies on different objects (distributions or random variables). The tightness of \( {{(\mathbb{P}_{X_i})}_{i\in I}} \) is strictly weaker than the UI of \( {{(X_i)}_{i\in I}} \).

UI functions with respect to a family of laws. The UI property for a family \( {{(X_i)}_{i\in I}\subset L^1} \) depends actually only on the marginal distributions \( {{(\mathbb{P}_{X_i})}_{i\in I}} \) and does not feel the dependence between the \( {X_i} \)’s. In this spirit, if \( {(\eta_i)_{i\in I}} \) is a family of probability measures on a Borel space \( {(E,\mathcal{E})} \) and if \( {f:E\rightarrow\mathbb{R}} \) is a Borel function then we say that \( {f} \) is UI for \( {(\eta_i)_{i\in I}} \) on \( {E} \) when

\[ \lim_{t\rightarrow\infty}\sup_{i\in I}\int_{\{|f|>t\}}\!|f|\,d\eta_i=0. \]

This means that \( {{(f(X_i))}_{i\in I}} \) is UI, where \( {X_i\sim\eta_i} \) for every \( {i\in I} \). This property is often used in applications as follows: if \( {\eta_n\rightarrow\eta} \) narrowly as \( {n\rightarrow\infty} \) for some probability measures \( {{(\eta_n)}_{n\geq1}} \) and \( {\eta} \) and if \( {f} \) is continuous and UI for \( {(\eta_n)_{n\geq1}} \) then

\[ \int\!|f|\,d\eta<\infty \quad\text{and}\quad \lim_{n\rightarrow\infty}\int\!f\,d\eta_n=\int\!f\,d\eta. \]

Logarithmic potential. What follows is extracted from the survey Around the circular law (random matrices). We have already devoted a previous post to the logarithmic potential. Let \( {\mathcal{P}(\mathbb{C})} \) be the set of probability measures on \( {\mathbb{C}} \) which integrate \( {\log\left|\cdot\right|} \) in a neighborhood of infinity. The logarithmic potential \( {U_\mu} \) of \( {\mu\in\mathcal{P}(\mathbb{C})} \) is the function \( {U_\mu:\mathbb{C}\rightarrow(-\infty,+\infty]} \) defined for all \( {z\in\mathbb{C}} \) by

\[ U_\mu(z)=-\int_{\mathbb{C}}\!\log|z-w|\,d\mu(w) =-(\log\left|\cdot\right|*\mu)(z). \]

Let \( {\mathcal{D}'(\mathbb{C})} \) be the set of Schwartz-Sobolev distributions on \( {\mathbb{C}} \). We have \( {\mathcal{P}(\mathbb{C})\subset\mathcal{D}'(\mathbb{C})} \). Since \( {\log\left|\cdot\right|} \) is Lebesgue locally integrable on \( {\mathbb{C}} \), the Fubini-Tonelli theorem implies that \( {U_\mu} \) is a Lebesgue locally integrable function on \( {\mathbb{C}} \). In particular, we have \( {U_\mu<\infty} \) almost everywhere and \( {U_\mu\in\mathcal{D}'(\mathbb{C})} \). By using Green’s or Stockes’ theorems, one may show, for instance via the Cauchy-Pompeiu formula, that for any smooth and compactly supported function \( {\varphi:\mathbb{C}\rightarrow\mathbb{R}} \),

\[ -\int_{\mathbb{C}}\!\Delta\varphi(z)\log|z|\,dxdy=2\pi\varphi(0) \]

where \( {z=x+iy} \). Now can be written, in \( {\mathcal{D}'(\mathbb{C})} \),

\[ \Delta\log\left|\cdot\right| = 2\pi\delta_0 \]

In other words, \( {\frac{1}{2\pi}\log\left|\cdot\right|} \) is the fundamental solution of the Laplace equation on \( {\mathbb{R}^2} \). Note that \( {\log\left|\cdot\right|} \) is harmonic on \( {\mathbb{C}\setminus\{0\}} \). It follows that in \( {\mathcal{D}'(\mathbb{C})} \),

\[ \Delta U_\mu=-2\pi\mu, \]

i.e. for every smooth and compactly supported “test function” \( {\varphi:\mathbb{C}\rightarrow\mathbb{R}} \),

\[ \langle\Delta U_\mu,\varphi\rangle_{\mathcal{D}’} =-\!\int_{\mathbb{C}}\!\Delta\varphi(z)U_\mu(z)\,dxdy =-2\pi\int_{\mathbb{C}}\!\varphi(z)\,d\mu(z) =-\langle2\pi\mu,\varphi\rangle_{\mathcal{D}’} \]

where \( {z=x+iy} \). Also \( {-\frac{1}{2\pi}U_\cdot} \) is the Green operator on \( {\mathbb{R}^2} \) (Laplacian inverse). For every \( {\mu,\nu\in\mathcal{P}(\mathbb{C})} \), we have

\[ U_\mu=U_\nu\text{ almost everywhere }\Rightarrow \mu=\nu. \]

To see it, since \( {U_\mu=U_\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \), we get \( {\Delta U_\mu=\Delta U_\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \), and thus \( {\mu=\nu} \) in \( {\mathcal{D}'(\mathbb{C})} \), and finally \( {\mu=\nu} \) as measures since \( {\mu} \) and \( {\nu} \) are Radon measures. (Note that this remains valid if \( {U_\mu=U_\nu+h} \) for some harmonic \( {h\in\mathcal{D}'(\mathbb{C})} \)). As for the Fourier transform, the pointwise convergence of logarithmic potentials along a sequence of probability measures implies the narrow convergence of the sequence to a probability measure. We need however some strong tightness. More precisely, if \( {{(\mu_n)}_{n\geq1}} \) is a sequence in \( {\mathcal{P}(\mathbb{C})} \) and if \( {U:\mathbb{C}\rightarrow(-\infty,+\infty]} \) is such that

  • (i) for a.a. \( {z\in\mathbb{C}} \), \( { \lim_{n\rightarrow\infty}U_{\mu_n}(z)=U(z)} \);
  • (ii) \( {\log(1+\left|\cdot\right|)} \) is UI for \( {(\mu_n)_{n \geq 1}} \);

then there exists \( {\mu \in \mathcal{P}(\mathbb{C})} \) such that

  • (j) \( {U_\mu=U} \) almost everywhere;
  • (jj) \( {\mu = -\frac{1}{2\pi}\Delta U} \) in \( {\mathcal{D}'(\mathbb{C})} \);
  • (jjj) \( {\mu_n\rightarrow\mu} \) narrowly.

Let us give a proof inspired from an article by Goldsheid and Khoruzhenko on random tridiagonal matrices. From the de la Vallée Poussin criterion, assumption (ii) implies that for every real number \( {r\geq1} \), there exists \( {\varphi\in\Phi} \), which may depend on \( {r} \), which is moreover convex and has moderate growth \( {\varphi(x)\leq 1+x^2} \), and

\[ \sup_n \int\!\varphi(\log(r+|w|))\,d\mu_{n}(w)<\infty. \]

Let \( {K\subset \mathbb{C}} \) be an arbitrary compact set. Take \( {r = r(K) \geq 1} \) large enough so that the ball of radius \( {r-1 } \) contains \( {K} \). Therefore for every \( {z\in K} \) and \( {w\in\mathbb{C}} \),

\[ \varphi(|\log|z-w||) \leq (1 + |\log|z-w||^2)\mathbf{1}_{\{|w|\leq r\}} +\varphi(\log(r+|w|))\mathbf{1}_{\{|w|>r\}}. \]

The couple of inequalities above, together with the local Lebesgue integrability of \( {(\log\left|\cdot\right|)^2} \) on \( {\mathbb{C}} \), imply, by using Jensen and Fubini-Tonelli theorems,

\[ \sup_n\int_K\!\varphi(|U_n(z)|)\,dxdy \leq \sup_n\iint\!\mathbf{1}_K(z)\varphi(|\log|z-w||)\,d\mu_n(w)\,dxdy<\infty, \]

where \( {z=x+iy} \) as usual. Since the de la Vallée Poussin criterion is necessary and sufficient for UI, this means that the sequence \( {{(U_{\mu_n})}_{n\geq1}} \) is locally Lebesgue UI. Consequently, from (i) it follows that \( {U} \) is locally Lebesgue integrable and that \( {U_{\mu_n}\rightarrow U} \) in \( {\mathcal{D}'(\mathbb{C})} \). Since the differential operator \( {\Delta} \) is continuous in \( {\mathcal{D}'(\mathbb{C})} \), we find that \( {\Delta U_{\mu_n}\rightarrow\Delta U} \) in \( {\mathcal{D}'(\mathbb{C})} \). Since \( {\Delta U\leq0} \), it follows that \( {\mu:=-\frac{1}{2\pi}\Delta U} \) is a measure (see e.g. Hormander). Since for a sequence of measures, convergence in \( {\mathcal{D}'(\mathbb{C})} \) implies narrow convergence, we get \( {\mu_n=-\frac{1}{2\pi}\Delta U_{\mu_n}\rightarrow\mu=-\frac{1}{2\pi}\Delta U} \) narrowly, which is (jj) and (jjj). Moreover, by assumptions (ii) we get additionally that \( {\mu\in\mathcal{P}(\mathbb{C})} \). It remains to show that \( {U_\mu=U} \) almost everywhere Indeed, for any smooth and compactly supported \( {\varphi:\mathbb{C}\rightarrow\mathbb{R}} \), since the function \( {\log\left|\cdot\right|} \) is locally Lebesgue integrable, the Fubini-Tonelli theorem gives

\[ \int\!\varphi(z)U_{\mu_n}(z)\,dz =-\int\!\left(\int\!\varphi(z)\log|z-w|\,dz\right)\,d\mu_n(w). \]

Now \( {\varphi*\log\left|\cdot\right|:w\in\mathbb{C}\mapsto\int\!\varphi(z)\log|z-w|\,dz} \) is continuous and is \( {\mathcal{O}(\log|1+\cdot|)} \). Therefore, by (i-ii), \( {U_{\mu_n}\rightarrow U_\mu} \) in \( {\mathcal{D}'(\mathbb{C})} \), thus \( {U_\mu=U} \) in \( {\mathcal{D}'(\mathbb{C})} \) and then almost everywhere, giving (j).

Last Updated on 2021-03-07

10 Comments

  1. Luc 2014-05-13

    Bonjour,

    In the proof of 2==>1 , you argued that” …. if \Omega = B_1 …. \cup B_n is a cover, then ….. ” But in general does such a cover exists ?

    Sincerely,

    Luc.

  2. Djalil Chafaï 2014-05-13

    you are right, something was missing: either one has to ask for the epsilon-delta for any probability space realizing (X_i), or one has to add the boundedness in L^1 to the epsilon-delta criterion. This is fixed now. Thanks.

  3. Luc 2014-08-13

    Bonjour,

    C’est moi encore.

    In the proof of 1 => 3, you select a strictly increasng sequecne $\{ x_m \}$, then construct $u_n$ and claimed that $\varphi\in\Phi$, since $u_n$ increases to infinity. I think here we need to clarify further about the choice of $x_m$. For example, the following construction fails:

    for m, choose $x_m$, an integer greater than $2^m$ such that $\sup_{i\in I} \mathbb{E}\Big[ \vert X_i \vert \cdot 1_{(\vert X_i \vert \geq x_m)} < 2^{-m}$,

    then pick another integer $x_{m+1} \geq (x_m + 1)\vee 2^{m+1}$, then

    $u_{2^n} \leq n$ so that

    $$\lim_{n\to+\infty} \frac{ u_1 + u_2 + \ldots + u_{2^n} }{2^n} = 0 $$
    and thus
    $$ \frac{\varphi(x)}{x} \xrightarrow{x\to+\infty} 0 \,\, . $$

  4. Luc 2014-08-13

    sorry, the counterexample is wrong!@Luc

  5. Djalil Chafaï 2014-08-13

    To be more explicit, since $\lim_{n\to\infty}u_n=+\infty$, we know that for every $A>0$ there exists $N=N_A$ such that $u_n\geq A$ for all $n\geq N$. Now, by definition of $\varphi$ we have $\varphi'(x)\geq A$ for all $x\geq N$, which gives $\varphi(x)/x\geq (1/x)\int_N^x\!\varphi'(t)\,dt\geq (1-N/x)A$ for any $x\geq N$. In particular $\varphi(x)/x\geq A/2$ for any $x\geq 2N$, and therefore $\lim_{x\to\infty}\varphi(x)/x=+\infty$.

  6. Luc 2014-08-14

    1.

    sorry again , the arithmetic mean $\dfrac{1}{n+1}\sum_{k=1}^n u_k \xrightarrow{n\to+\infty} +\infty $. Here is an elementary proof:

    For any $M > 0$, there exists $N_1\in\mathbb{N}^\ast$ such that $u_n > M $ as $n\geq N_1$

    and there exists $N_2\in\mathbb{N}^\ast$ such that $N_2 > N_1$ and $u_n > 2M$, as $n \geq N_2$,

    then for $n > N_1 + N_2 + 1$, we have
    \begin{align*}
    \frac{u_1 + u_2 + \ldots + u_n }{n+1} &= \frac{u_1 + u_2 + \ldots + u_{N_1} }{n+1} + \frac{u_{N_1 + 1} + \ldots + u_{N_2} }{n+1} + \frac{u_{N_2 + 1} + \ldots + u_n }{n+1} \\
    & \geq \frac{u_{N_1 + 1} + \ldots + u_{N_2} }{n+1} + \frac{u_{N_2 + 1} + \ldots + u_n }{n+1} \\
    & \geq \frac{(N_2 – N_1)\times M}{n+1} + \frac{(n – N_2)\times (2M)}{n+1} \\
    & = 2M – \frac{2 + N_2 + N_1}{n+1}\cdot M > M \,\, .
    \end{align*}
    This implies $\dfrac{\varphi(n+1)}{n+1} \xrightarrow{n\to+\infty} +\infty$ hence $\dfrac{\varphi(x)}{x} \geq \dfrac{\varphi( \lfloor x \rfloor)}{ \lfloor x \rfloor } \cdot \dfrac{ x- 1 }{x} \xrightarrow{x\to+\infty} +\infty$.

    2.

    In the proof of $\Big( 1\Longrightarrow 3 \Big)$ , the very last equality should be changed to the inequality $\leq$ ? see the following ?

    \begin{align*}
    &\ldots\ldots = \sum_{m\geq 1} \sum_{n\geq x_m} \mathbb{P} \big( \vert X_i \vert \geq n \big) \\
    & = \sum_{m=1}^\infty \left[ \,\, \sum_{n \geq x_m} \sum_{k=n}^\infty \mathbb{P} \big( k \leq \vert X_i \vert < k+1 \big) \,\, \right] \\
    & = \sum_{m=1}^\infty \left[ \,\, \sum_{k \geq x_m} \sum_{n = x_m}^k \mathbb{P} \big( k \leq \vert X_i \vert < k+1 \big) \,\, \right] \\
    & = \sum_{m=1}^\infty \sum_{k \geq x_m} (k – x_m) \mathbb{P} \big( k \leq \vert X_i \vert < k+1 \big) \\
    & \leq \sum_{m=1}^\infty \sum_{k \geq x_m} k \cdot \mathbb{P} \big( k \leq \vert X_i \vert < k+1 \big)
    \end{align*}

  7. Djalil Chafaï 2014-08-14

    1. All right. 2. Youare right, and this glitch is now fixed, thanks.

  8. Djalil Chafaï 2018-11-10

    For French readers, the concept of uniform integrability, under the name équi-intégrabilité, is considered in the book « Bases mathématiques du calcul des probabilités » by Jacques Neveu. In particular the de la Vallée Poussin criterion is considered, sligltly and without the name and the equivalence, in Exercice II-5-2 page 52. This precise reference was communicated by Laurent Miclo. For a more substantial treatment, the best remains the book by Claude Delacherie and Paul-André Meyer.

  9. Xa 2021-03-05

    Bonjour,

    It is not true that the span of a family of UI is UI, the span of 1 (i.e. R) is not UI. Is it correct?

    Merci

  10. Djalil Chafaï 2021-03-06

    Thank you very much, you are perfectly right. It does not work with the span. Actually it is a matter of tightness, because it works nicely with the convex envelope. I have updated the post accordingly.

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