# Libres pensées d'un mathématicien ordinaire Posts

This post gives below the Mathematical Citation Quotient (MCQ) from 2000 to 2018 for journals in probability, statistics, analysis, and general mathematics. The numbers were obtained using home brewed scripts and MathSciNet data. The graphics were created with LibreOffice.

Recall that the MCQ is a ratio of two counts for a selected journal and a selected year.  The MCQ for year $Y$ and journal $J$ is given by the formula $\mathrm{MCQ}=m/n$ where

• $m$ is the total number of citations of papers published in jounal $J$ in years $Y-1$,…,$Y-5$ by papers published in year $Y$ in any journal known by MathSciNet;
• $n$ is the total number of papers published in journal $J$ in years $Y-1$,…,$Y-5$.

The Mathematical Reviews compute every year the MCQ for every indexed journal, and make it available on MathSciNet. This formula is very similar to the one of the five years impact factor, the main difference being the population of journals which is specifically mathematical for the MCQ (reference list journals) and the way the citations are extracted. Both biases are negative.

The MCQ is a rough measurement of the social scientific value of journals. The results are quite compatible with what we have in mind. The trends are sometimes intriguing. For probability journals, for instance, it seems that there are three groups. This reminds reinforcement or self-organized criticality. The first group is AOP-PTRF-CMP-JFA-AAP, with some hesitations and an “Annals” naming effect. The second group is AIHP-EJP-SPA-Bernoulli, the third group is ALEA-ECP-AdAP-JAP-JTP-ESAIM. We observe some transitions from one group to another, for instance, since 2010, AAP moved to the first group while ALEA moved to the third group. The case of ECP is very special since its papers are half the standard size. The MCQ probably underestimates the social value of ECP by a rough factor 2, which is logical if we compare with EJP.

Yes, there are more robust ways to measure the social value of a journal, such as for instance the (recursive) eigenfactor, and it could be interesting to check if the three groups are stable!    Σ

Let $X=(X_1,\ldots,X_n)$ be a random vector of $(\mathbb{R}^d)^n$ with density proportional to $$(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n\mapsto\mathrm{e}^{-\beta\sum_{i=1}^nV(x_i)}\prod_{i<j}W(x_i-x_j),$$ where $V,W:\mathbb{R}^d\to\mathbb{R}$ are homogeneous functions, with $W\geq0$. This means that there exist $a,b\geq0$ such that for all $\lambda\geq0$ and $x\in\mathbb{R}^d$, $V(\lambda x)=\lambda^a V(x)$ and $W(\lambda x)=\lambda^bW(x)$. Now, for all $\theta>0$, by the change of variable $x_i=\sqrt[a]{\beta/(\theta+\beta)}y_i$,
\begin{multline*}
\int_{(\mathbb{R}^d)^n}\mathrm{e}^{-(\theta+\beta)\sum_iV(x_i)}\prod_{i<j}W(x_i-x_j)\mathrm{d}x\\
=\Bigr(\frac{\beta}{\theta+\beta}\Bigr)^{\frac{nd}{a}+\frac{n(n-1)a}{2b}}
\int_{(\mathbb{R}^d)^n}\mathrm{e}^{-\beta\sum_iV(y_i)}\prod_{i<j}W(y_i-y_j)\mathrm{d}y.
\end{multline*}
We recognize the Laplace transform of a Gamma distribution, since
$\int_0^\infty\mathrm{e}^{-\theta u}u^{\alpha-1}\mathrm{e}^{-\beta u}\mathrm{d}u =\int_0^\infty u^{\alpha-1}\mathrm{e}^{-(\theta+\beta)u}\mathrm{d}u =\Bigr(\frac{\beta}{\theta+\beta}\Bigr)^\alpha\frac{\Gamma(\alpha)}{\beta^\alpha},$and we obtain
$\sum_iV(X_i)\sim\mathrm{Gamma}\Bigr(\frac{nd}{a}+\frac{n(n-1)bd}{2a},\beta\Bigr).$
A remarkable general fact! The case $V=\frac{1}{2}\left|\cdot\right|^2$ and $W=\left|\cdot\right|^\beta$ corresponds to the beta Ginibre gas of random matrix theory. The case $V=\frac{n+1}{2}\log(1+\left|\cdot\right|^2)$ and $W=\left|\cdot\right|^2$ corresponds to the Forrester–Krishnapur spherical gas of random matrix theory.

We could generalize even more,  and replace $(x_1,\ldots,x_n)\mapsto\sum_iV(x_i)$ by a homogenenous $(x_1,\ldots,x_n)\mapsto V(x_1,\ldots,x_n)$ and $(x_1,\ldots,x_n)\mapsto\prod_{i<j}W(x_i-x_j)$ by a homogeneous $(x_1,\ldots,x_n)\mapsto W(x_1,\ldots,x_n)$, in the sense that for some $a,b\geq0$ and all $\lambda\geq0$, $x\in(\mathbb{R}^d)^n$, $V(\lambda x)=\lambda^aV(x)$ and $W(\lambda x)=\lambda^bW(x)$. In this case $X=(X_1,\ldots,X_n)$ has density proportional to $x\in(\mathbb{R}^d)^n\mapsto\mathrm{e}^{-\beta V(x)}W(x)$. This would hide the structure of exchangeable gas with pair-interaction that we had in mind for the examples. But this would give $$V(X)=V(X_1,\ldots,X_n)\sim\mathrm{Gamma}\Bigr((n+b)\frac{d}{a},\beta\Bigr).$$

Ω

Suppose that we would like to describe mathematically the convergence of a sequence ${(X_n)}_n$ of random variables towards a limiting random variable $X_\infty$, as $n\to\infty$. We have to select a notion of convergence. If we decide to use almost sure convergence, we need to define all the $X_n$’s as well as the limit $X_\infty$ on a common probability space in order to give a meaning to $$\mathbb{P}(\lim_{n\to\infty}X_n=X_\infty)=1.$$ This means that we need to couple the random variables. If we decide to use convergence in probability or in $L^p$, we have to define, for all $n$, both $X_n$ and $X_\infty$ in the same probability space in order to give a meaning to $\mathbb{P}(|X_n-X_\infty|>\varepsilon)$ and $\mathbb{E}(|X_n-X_\infty|^p)$ respectively, and therefore we end up to define all the $X_n$’s as well as $X_\infty$ on a common probability space. However, if we decide to use convergence in law (i.e. in distribution), then we do not need at all to define the random variables on a common probability space.

In the special case where $X_\infty$ is deterministic, the convergence in probability or in $L^p$ no longer impose to define the random variables on the same probability space. However, the almost sure convergence still requires the same probability space. Moreover if we impose that the almost sure convergence holds regardless of the way we define the random variables on the same probability space (i.e. for arbitrary couplings), then we end up with the important notion of complete convergence, which is equivalent, thanks to Borel-Cantelli lemmas, to a summable convergence in probability. Note that when the limit is deterministic, we also know that the convergence in law is equivalent to the convergence in probability. Moreover, we know in general from the Borel-Cantelli lemma that a summable convergence in probability implies almost sure convergence. Furthermore, the convergence in probability becomes easily summable under moment conditions.

Following Hsu & Robbins, if we consider $X_n=\frac{1}{n}(Z_1+\cdots+Z_n)$ where $Z_1,\ldots,Z_n$ are independent copies of some $Z$ of mean $m$, then the sequence ${(X_n)}_n$ converges completely towards $m$ as soon as $Z$ has a finite second moment, and this condition is almost necessary. This sheds an interesting light on the law of large numbers for triangular arrays.

Some people refuse to consider the almost sure convergence as a true mode of convergence in the sense that it is not associated to a metric, contrary to the other modes of convergence. In some sense, it appears as a critical notion in the law of large numbers, when we lower the concentration typically via integrability (moments conditions). Of course there are plenty of concrete situations for instance with martingales in which the coupling is in fact imposed and for which the almost sure convergence towards a non-constant random variable holds very naturally. A famous example is for instance the one of Pólya urns and of Galton-Watson branching processes. The Marchenko-Pastur theorem in random matrix theory provides an example of natural coupling with a limiting object which is deteterministic, and the convergence is complete via concentration of measure provided that the ingredients have enough finite moments.

Note. The idea of writing this tiny post came from a discussion with my friend Adrien Hardy.

Recently, during a coffee break, emerged a discussion about the presence of probability and statistics in top journals such as Annals of mathematics, Acta Mathematica, Inventiones Mathematicae, or Journal of the AMS. Well, the question has an interest from the point of view of the sociology and history of science. Let us use the Primary and Secondary Mathematical Subject Classification (MSC) codes of each article in order to detect Probability (60x) or Statistics (62x). Here is the data from MathSciNet/zbMath:

• Annals of Mathematics published 4464 papers in total from 1938 to 2019.
Among them, 76 (1.7%) have Primary MSC 60x [PDF]
Among them, 112 (2.5%) have Primary or Secondary MSC 60x [PDF]
Moreover only 2 have Primary or Secondary MSC 62x [PDF]
• Acta Mathematica published 1297 papers in total from 1938 to 2017.
Among them, 44 (3.4%) have Primary MSC 60x [PDF]
Among them, 63 (4.9%) have Primary or Secondary MSC 60x [PDF]
Moreover only 4 have Primary or Secondary MSC 62x [PDF]
• Inventiones Mathematicae published 4311 papers in total from 1966 to 2019.
Among them, 52 (1.2%) have Primary MSC 60x [PDF]
Among them, 95 (2.2%) have Primary or Secondary MSC 60x [PDF]
Moreover only 2 have Primary or Secondary MSC 62x [PDF]
• Journal of the AMS published 963 papers in total from 1988 to 2019.
Among them, 28 (2.9%) have Primary MSC 60x [PDF]
Among them, 49 (5.1%) have Primary or Secondary MSC 60x [PDF]
Moreover only 5 have Primary or Secondary MSC 62x [PDF]

The presence of probability is low, while the one of statistics is microscopic. A scandal.

AO(P|S). Annals of Probability (AOP) and Annals of Statistics (AOS) were founded only in 1973.

1938. Annals of Mathematics is historically American whereas Acta Mathematica is European. They started respectively in 1892 and 1882. According to MathSciNet, it seems that the first article classified 60x in these journals was published in 1938. The MSC by itself was introduced at the end of the thirties and many articles in MathSciNet are not classified before 1940 at the time of writing. Note that N. Wiener published in the twenties while A. N. Kolmogorov published in the thirties.

Why. The phenomenon has probably multiple explanations, among them we could mention for instance the possible effects of utilitarism and anti-utilitarism in the mathematical elite, in particular during the fifties and sixties, and the possible overweight of some kind of “snobish pure mathematics or mathematicians” in top journals boards. We could also see AOP and AOS as some sort of mathematical ghettos and think about self-censorship. We could moreover think about generational effects. Finally we have to keep in mind that some probability papers were published without any primary or secondary 60x code, such as for instance this one  or that one.

Here is some additional data provided by MathSciNet for Annals of Mathematics:

MSCDescriptionCount
Other (includes unclassified papers before 1940)1626
14Algebraic geometry319
57Manifolds and cell complexes298
20Group theory and generalizations292
11Number theory288
53Differential geometry262
46Functional analysis207
58Global analysis, analysis on manifolds203
56Other200
32Several complex variables and analytic spaces174
55Algebraic topology172
10Number theory159
22Topological groups, Lie groups149
30Functions of a complex variable135
35Partial differential equations126
42Harmonic analysis on Euclidean spaces111
37Dynamical systems and ergodic theory107
09Other100
60Probability theory and stochastic processes76
54General topology51
27Other47
36Other43
12Field theory and polynomials40
49Calculus of variations and optimal control; optimization39
17Nonassociative rings and algebras35
47Operator theory35
02Logic and foundations33
05Combinatorics32
52Convex and discrete geometry32
34Ordinary differential equations31
81Quantum theory26
28Measure and integration25
16Associative rings and algebras22
03Mathematical logic and foundations21
13Commutative algebra20
40Sequences, series, summability20
18Category theory; homological algebra19
83Relativity and gravitational theory19
31Potential theory18
43Abstract harmonic analysis17
19K-theory13
82Statistical mechanics, structure of matter13
90Operations research, mathematical programming12
06Order, lattices, ordered algebraic structures11
41Approximations and expansions11
15Linear and multilinear algebra; matrix theory10
33Special functions10
26Real functions9
48Other8
45Integral equations7
76Fluid mechanics7
04Set theory6
39Difference and functional equations6
70Mechanics of particles and systems6
00General3
68Computer science3
44Integral transforms, operational calculus2
62Statistics2
73Mechanics of solids2
01History and biography1
21Other1
71Other1
79Other1
80Classical thermodynamics, heat transfer1
84Other1
94Information and communication, circuits1

The same for the last three years :

MSCDescriptionCount
11Number theory29
14Algebraic geometry20
53Differential geometry17
05Combinatorics10
35Partial differential equations10
37Dynamical systems and ergodic theory8
20Group theory and generalizations6
58Global analysis, analysis on manifolds6
03Mathematical logic and foundations5
57Manifolds and cell complexes5
13Commutative algebra4
22Topological groups, Lie groups4
32Several complex variables and analytic spaces4
60Probability theory and stochastic processes4
42Harmonic analysis on Euclidean spaces3
49Calculus of variations and optimal control; optimization3
52Convex and discrete geometry3
55Algebraic topology3
28Measure and integration2
30Functions of a complex variable2
46Functional analysis2
83Relativity and gravitational theory2
06Order, lattices, ordered algebraic structures1
19K-theory1
43Abstract harmonic analysis1
47Operator theory1

JMPA. We could think that a journal such as Journal de mathématiques pures et appliquées, founded in 1872, is in the same time relatively prestigious, generalist, and more open to applied mathematics in general and to probability and statistics in particular. Here is the data for all MSC codes, taken from MathSciNet. We see an obvious overweight for partial differential equations. In the mean time, the situation of probability is better than before, while the presence of statistics is still microscopic.

MSCDescriptionCount
35Partial differential equations801
58Global analysis, analysis on manifolds138
53Differential geometry136
49Calculus of variations and optimal control; optimization107
46Functional analysis86
76Fluid mechanics69
14Algebraic geometry61
60Probability theory and stochastic processes60
93Systems theory; control58
30Functions of a complex variable57
47Operator theory55
32Several complex variables and analytic spaces53
34Ordinary differential equations43
37Dynamical systems and ergodic theory42
42Harmonic analysis on Euclidean spaces35
74Mechanics of deformable solids34
31Potential theory32
81Quantum theory24
82Statistical mechanics, structure of matter23
22Topological groups, Lie groups21
73Mechanics of solids21
20Group theory and generalizations19
83Relativity and gravitational theory19
36Other17
11Number theory16
45Integral equations16
28Measure and integration15
56Other12
10Number theory11
17Nonassociative rings and algebras10
26Real functions10
54General topology10
65Numerical analysis10
78Optics, electromagnetic theory10
27Other9
33Special functions9
48Other8
50Geometry8
92Biology and other natural sciences8
05Combinatorics7
43Abstract harmonic analysis7
91Game theory, economics, social and behavioral sciences7
09Other6
55Algebraic topology6
02Logic and foundations5
12Field theory and polynomials5
15Linear and multilinear algebra; matrix theory5
41Approximations and expansions5
16Associative rings and algebras4
44Integral transforms, operational calculus4
52Convex and discrete geometry4
57Manifolds and cell complexes4
62Statistics4
70Mechanics of particles and systems4
40Sequences, series, summability3
85Astronomy and astrophysics3
13Commutative algebra2
39Difference and functional equations2
71Other2
80Classical thermodynamics, heat transfer2
86Geophysics2
90Operations research, mathematical programming2
94Information and communication, circuits2
18Category theory; homological algebra1
79Other1

The same for the last three years:

MSCDescriptionCount
35Partial differential equations154
49Calculus of variations and optimal control; optimization25
53Differential geometry19
14Algebraic geometry16
93Systems theory; control16
58Global analysis, analysis on manifolds12
42Harmonic analysis on Euclidean spaces8
37Dynamical systems and ergodic theory7
60Probability theory and stochastic processes7
76Fluid mechanics7
31Potential theory6
32Several complex variables and analytic spaces6
34Ordinary differential equations5
46Functional analysis5
81Quantum theory5
11Number theory4
47Operator theory4
74Mechanics of deformable solids4
28Measure and integration2
30Functions of a complex variable2
43Abstract harmonic analysis2
82Statistical mechanics, structure of matter2
83Relativity and gravitational theory2
92Biology and other natural sciences2
05Combinatorics1
12Field theory and polynomials1
15Linear and multilinear algebra; matrix theory1
20Group theory and generalizations1
26Real functions1
45Integral equations1
65Numerical analysis1
70Mechanics of particles and systems1
78Optics, electromagnetic theory1
86Geophysics1
91Game theory, economics, social and behavioral sciences1
94Information and communication, circuits1

CPAM. Finally, here is the same data for Communication on Pure and Applied Mathematics. This journal, established in 1948, is truly open to applied mathematics in general and to probability theory in particular. However, the presence of statistics is still extremely low.

MSCDescriptionCount
35Partial differential equations898
76Fluid mechanics234
58Global analysis, analysis on manifolds182
60Probability theory and stochastic processes177
53Differential geometry97
65Numerical analysis92
82Statistical mechanics, structure of matter92
34Ordinary differential equations85
47Operator theory65
49Calculus of variations and optimal control; optimization64
37Dynamical systems and ergodic theory58
78Optics, electromagnetic theory58
20Group theory and generalizations49
46Functional analysis48
81Quantum theory43
10Number theory39
73Mechanics of solids37
30Functions of a complex variable29
36Other25
32Several complex variables and analytic spaces24
57Manifolds and cell complexes24
11Number theory23
74Mechanics of deformable solids23
94Information and communication, circuits22
42Harmonic analysis on Euclidean spaces20
03Mathematical logic and foundations15
31Potential theory15
45Integral equations15
62Statistics15
14Algebraic geometry14
55Algebraic topology14
70Mechanics of particles and systems14
83Relativity and gravitational theory14
92Biology and other natural sciences14
01History and biography13
15Linear and multilinear algebra; matrix theory13
52Convex and discrete geometry13
44Integral transforms, operational calculus12
22Topological groups, Lie groups11
26Real functions9
80Classical thermodynamics, heat transfer9
85Astronomy and astrophysics8
86Geophysics8
05Combinatorics7
43Abstract harmonic analysis7
12Field theory and polynomials6
28Measure and integration6
90Operations research, mathematical programming6
00General5
39Difference and functional equations5
68Computer science5
93Systems theory; control5
41Approximations and expansions4
91Game theory, economics, social and behavioral sciences4
02Logic and foundations3
17Nonassociative rings and algebras3
33Special functions3
09Other2
16Associative rings and algebras2
27Other2
40Sequences, series, summability2
54General topology2
69Other2
19K-theory1
51Geometry1
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