# Frédéric OuimetMcGill University | McGill · Department of Mathematics and Statistics

Frédéric Ouimet

PhD in Mathematics (2019) - University of Montreal

## About

114

Publications

13,072

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503

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Introduction

I completed my PhD in mathematics at Université de Montréal in May 2019 under Louis-Pierre Arguin and Alexander Fribergh. From September 2019 to May 2021, I was a postdoctoral scholar at Caltech under Maksym Radziwill. From June 2021 to August 2022, I was a postdoc at McGill University under Christian Genest. From September 2022 to August 2023, I was a CRM-Simons postdoc at Université de Montréal under Christian Genest. I am currently a postdoc at McGill University under Christian Genest.

Additional affiliations

September 2023 - present

September 2022 - August 2023

**Université de Montréal**

Position

- CRM-Simons Postdoctoral Fellow

June 2021 - August 2022

**McGill University**

Position

- Postdoctoral Researcher

## Publications

Publications (114)

We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. T...

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\oo(T)$, \begin{equation*} \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}} |\zeta(\tfrac 12 + \ii t + \ii h)|^{\beta} \rd h = (\log T)^{f_{\theta}(\beta) + \oo(1)}, \end{equation*} for some explicit exponent $f_{\theta}(\beta)$, where $\theta > -1$ and $\beta > 0$. This...

This paper presents a novel approach for pointwise estimation of multivariate density functions on known domains of arbitrary dimensions using nonparametric local polynomial estimators. Our method is highly flexible, as it applies to both simple domains, such as open connected sets, and more complicated domains that are not star-shaped around the p...

This paper extends various results related to the Gaussian product inequality (GPI) conjecture to the setting of disjoint principal minors of Wishart random matrices. This includes product-type inequalities for matrix-variate analogs of completely monotone functions and Bernstein functions of Wishart disjoint principal minors, respectively. In part...

This paper introduces a novel density estimator supported on d-dimensional half-spaces. It stands out as the first asymmetric kernel smoother for half-spaces in the literature. Using the multivariate inverse Gaussian (MIG) density from Minami (2003) as the kernel and incorporating locally adaptive parameters, the estimator achieves desirable bounda...

This paper extends various results related to the Gaussian product inequality (GPI) conjecture to the setting of disjoint principal minors of Wishart random matrices. This includes product-type inequalities for matrix-variate analogs of completely monotone functions and Bernstein functions of Wishart disjoint principal minors, respectively. In part...

This paper provides the first explicit formula for the expectation of the product of two disjoint principal minors of a Wishart random matrix, solving a part of a broader problem put forth by Samuel S. Wilks in 1934 in the Annals of Mathematics. The proof makes crucial use of hypergeometric functions of matrix argument and their Laplace transforms....

This paper provides the first explicit formula for the expectation of the product of two disjoint principal minors of a Wishart random matrix, solving a part of a broader problem put forth by Samuel S. Wilks in 1934 in the Annals of Mathematics. The proof makes crucial use of hypergeometric functions of matrix argument and their Laplace transforms....

The large-sample behavior of non-degenerate multivariate U-statistics of arbitrary degree is investigated under the assumption that their kernel depends on parameters that can be estimated consistently. Mild regularity conditions are given which guarantee that once properly normalized, such statistics are asymptotically multivariate Gaussian both u...

The large-sample behavior of non-degenerate multivariate U-statistics of arbitrary degree is investigated under the assumption that their kernel depends on parameters that can be estimated consistently. Mild regularity conditions are given which guarantee that once properly normalized, such statistics are asymptotically multivariate Gaussian both u...

This paper presents a novel approach for pointwise estimation of multivariate density functions on known domains of arbitrary dimensions using nonparametric local polynomial estimators. Our method is highly flexible, as it applies to both simple domains, such as open connected sets, and more complicated domains that are not star-shaped around the p...

This paper introduces a local linear smoother for regression surfaces on the simplex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring excellent boundary properties. Asymptotic results for the bias, variance, mean squared error, and mean integrated squared error are derived, generaliz...

This paper introduces a local linear smoother for regression surfaces on the sim-plex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring excellent boundary properties. Asymptotic results for the bias, variance, mean squared error, and mean integrated squared error are derived, generali...

This paper presents a multivariate normal integral expression for the joint survival function of the cumulated components of any multinomial random vector. This result can be viewed as a multivariate analog of Equation (7) from Carter and Pollard [2], who improved Tusnády's inequality. Our findings are based on a crucial relationship between the jo...

This paper presents a multivariate normal integral expression for the joint survival function of the cumulated components of any multinomial random vector. This result can be viewed as a multivariate analog of Equation (7) from Carter & Pollard (2004), who improved Tusn\'ady's inequality. Our findings are based on a crucial relationship between the...

In 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but litt...

This paper introduces a novel density estimator supported on d-dimensional half-spaces. It stands out as the first asymmetric kernel smoother for half-spaces in the literature. Using the multivariate inverse Gaussian (MIG) density from Minami (2003) as the kernel and incorporating locally adaptive parameters, the estimator achieves desirable bounda...

This paper introduces a novel density estimator supported on d-dimensional half-spaces. It stands out as the first asymmetric kernel smoother for half-spaces in the literature. Using the multivariate inverse Gaussian (MIG) density from Minami (2003) as the kernel and incorporating locally adaptive parameters, the estimator achieves desirable bounda...

In 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but litt...

In 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but litt...

In this short note, explicit formulas are developed for the central and noncentral moments of the multi-variate hypergeometric distribution. A numerical implementation is provided in Mathematica for fast evaluations. This work complements the paper by Ouimet [4], where analogous formulas were derived and implemented in Mathematica for the multinomi...

In this short note, explicit formulas are developed for the central and noncentral moments of the multi-variate hypergeometric distribution. A numerical implementation is provided in Mathematica for fast evaluations. This work complements the paper by Ouimet [4], where analogous formulas were derived and implemented in Mathematica for the multinomi...

Dans cette présentation, nous rappellerons quelques définitions et résultats préliminaires sur l'analyse matricielle dans un contexte statistique. Nous procéderons ensuite à une revisite de certaines formules relativement méconnues développées par le statisticien Samuel S. Wilks en 1934, concernant les moments joints de mineurs principaux, imbriqué...

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of ‘failures’ occur in a finite setting, the NM model is typically chosen over the more ac...

In this short note, we present a refined approximation for the log-ratio of the density of the von Mises(µ, κ) distribution (also called the circular normal distribution) to the standard (linear) normal distribution when the concentration parameter κ is large. Our work complements the one of Hill (1976), who obtained a very similar approximation al...

In this short note, we present a refined approximation for the log-ratio of the density of the von Mises(µ, κ) distribution (also called the circular normal distribution) to the standard (linear) normal distribution when the concentration parameter κ is large. Our work complements the one of Hill (1976), who obtained a very similar approximation al...

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of 'failures' occur in a finite setting, the NM model is typically chosen over the more ac...

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of 'failures' occur in a finite setting, the NM model is typically chosen over the more ac...

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper b...

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper b...

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper b...

In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a compar...

In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a compar...

In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a compar...

In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to...

In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to...

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, Mosimann (1963) derived general formulas for the falling factorial moments of the negative multinomial distribution, while Withers & Nadarajah (2014) obtained expressions...

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, Mosimann (1963) derived general formulas for the falling factorial moments of the negative multinomial distribution, while Withers & Nadarajah (2014) obtained expressions...

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a NegativeBinomial(r,p) random variable jittered by a Uniform(0,1), which answers a problem left open in Coeurjolly & Trépanier (2020)....

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a NegativeBinomial(r,p) random variable jittered by a Uniform(0,1), which answers a problem left open in Coeurjolly & Trépanier (2020)....

This paper considers the asymptotic behavior in β-Hölder spaces, and under Lp losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a c...

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing t...

This paper is concerned with the asymptotic behavior in β-Hölder spaces and under Lp losses of a Dirichlet kernel density estimator proposed by Aitchison & Lauder (1985) for the analysis of compositional data. In recent work, Ouimet & Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this nonparametric e...

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. Our results generalize those found by Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (...

In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the d-dimensional simplex. Our results generalize those found by Leblanc (2012), who treated the case d=1, and complement the results from Ouimet (2021...

In this paper we present the results from an empirical power comparison of 40 goodness-of-fit tests for the univariate Laplace distribution, carried out using Monte Carlo simulations with sample sizes n=20,50,100,200, significance levels α=0.01,0.05,0.10, and 400 alternatives consisting of asymmetric and symmetric light/heavy-tailed distributions t...

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kerne...

Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests...

Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests...

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X=(X1,…,Xd) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X is shown to be...

In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures....

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X=(X1,…,Xd) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X is shown to be...

In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures....

In this paper we present the results from an empirical power comparison of 40 goodness-of-fit tests for the univariate Laplace distribution, carried out using Monte Carlo simulations with sample sizes n=20,50,100,200, significance levels α=0.01,0.05,0.10, and 400 alternatives consisting of asymmetric and symmetric light/heavy-tailed distributions t...

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei [32] and Liu et al. [15] are extended in two ways: by replacing the power functions with more general classes of functions and by replacing the us...

In this paper, we prove a local limit theorem for the chi-square distribution with r > 0 degrees of freedom and noncentrality parameter λ ≥ 0. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of r −2 , which is significantly smaller than the maximal error bounds of order r −1/2 rec...

In this paper, we prove a local limit theorem for the chi-square distribution with $r > 0$ degrees of freedom and noncentrality parameter $\lambda \geq 0$. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of $r^{-2}$, which is significantly smaller than the maximal error bounds of...

In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points...

In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points...

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distributio...

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kerne...

We prove a non-asymptotic generalization of the refined continuity correction for the Binomial distribution found in Cressie (1978), which we then use to improve the versions of Tusnády's inequality from Massart (2002) and Carter & Pollard (2004) in the bulk.

In this paper, we introduce a matrix-parametrized generalization of the multinomial probability mass function and we show that it is logarithmically completely monotonic. New combinatorial inequalities involving ratios of multivariate gamma functions are derived from this result.

In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmically complete monotonicity of this generalization and derive new inequalities involving ratios of multivariate gamma functions.

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter (2002) and Ouimet (2021) on the total variation between the law of a multinomial vector jittered by a uniform on...

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter [4] and Ouimet [14] on the total variation between the law of a multinomial vector jittered by a uniform on (−1/2...

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distributio...

We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with known category-based values. For small sample sizes, we show how to leverage knowledge of the set of possible valu...

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F)....

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F)....

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N⁻¹. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how...

We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with known category-based values. For small sample sizes, we show how to leverage knowledge of the set of possible valu...

We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with known category-based values. For small sample sizes, we show how to leverage knowledge of the set of possible valu...

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than...

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line [0, ∞). They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than tradi...

We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. T...

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when d=2. The simplex is an...

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives...

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when d=2. The simplex is an...

In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N^{−1}. In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show ho...

We prove a non-asymptotic generalization of the refined continuity correction for the Binomial distribution found in Cressie (1978), which we then use to improve the versions of Tusn\'ady's inequality from Massart (2002) and Carter & Pollard (2004) in the bulk.

In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives...

The classical Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of n mutually independent and identically distributed random variables with finite second moment converges in distribution to a standard Gaussian as n goes to infinity. In particular, pairwise ind...

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. Thes...

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. Thes...

We give the first explicit formulas for the joint third and fourth central moments of the multinomial distribution, by differentiating the moment generating function. A general formula for the joint factorial moments was previously given in Mosimann (1962).

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of n mutually independent and identically distributed random variables with finite first and second moments converges in distribution to a standard Gaussian as n goes to infinity. In particular, pairwise in...

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N−1. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how...

We give the first explicit formulas for the joint third and fourth central moments of the multinomial distribution, by differentiating the moment generating function. A general formula for the joint factorial moments was previously given in Mosimann (1962).

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally...

In this short note, we study a random field that approximates the real part of the logarithm of the Riemann zeta function on the critical line, introduced by Harper (2013); Arguin et al. (2017), and its derivatives of all orders. We show that the maximum of the random field, and more generally the maximum of its j-th derivative, varies on a (log T)...

Let $m,n\in \mathbb{N}$. For all $i\in \{1,2,\dots,m\}$, $j\in \{1,2,\dots,n\}$, choose constants $X_i,Y_j > 0$, $M\in (0,1]$ and let $(x_{ij})\in (0,\infty)^{m\times n}$ be such that $\sum_{j=1}^n x_{ij} = X_i$, $\sum_{i=1}^m x_{ij} = Y_j$ and $\sum_{i=1}^m X_i = \sum_{j=1}^n Y_j = M$. We prove that the ratio of gamma functions \begin{equation*} a...

Final version of my thesis (accepted).
425 pages.

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good mod...

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\oo(T)$,
\begin{equation*}
\int_{-(\log T)^{\theta}}^{(\log T)^{\theta}} |\zeta(\tfrac 12 + \ii t + \ii h)|^{\beta} \rd h = (\log T)^{f_{\theta}(\beta) + \oo(1)},
\end{equation*}
for some explicit exponent $f_{\theta}(\beta)$, where $\theta > -1$ and $\beta > 0$. This...

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good mod...

## Questions

Questions (2)

Is there a multivariate analogue to the Bernstein-Widder theorem which says that a function on $[0,\infty)$ is completely monotonic if and only if it is the Laplace transform of a non-negative measure ?