A general decoupling inequality for finite Gaussian vectors
We use Matrix Analysis to prove a general decoupling inequality for finite Gaussian vectors, in identifying a new region of the inherent $p$ exponent, for the validity of this one.
math.PR↗
arXiv subjects
Michel Weber
Publications and source records attributed to Michel Weber.
Moderate deviations of suprema of Gaussian processes A cyclic approximation criterion
We study moderate deviations of suprema of parametrized sequences of sample bounded Gaussian processes $\{X _x(t), t\in T _x\}$, and first present recent sharp bounds in simple cases. In the almost periodic case, we prove an approximation theorem. We introduce a modulable diophantine approximation. Finally we study for general non-vanishing coefficient sequences, the behavior along lattices of almost periodic Gaussian polynomials with linearly independent frequencies, and use a lattice localized version of Kronecker's theorem.
math.PR↗
Sidekick compilation with xDSL
Traditionally, compiler researchers either conduct experiments within an existing production compiler or develop their own prototype compiler; both options come with trade-offs. On one hand, prototyping in a production compiler can be cumbersome, as they are often optimized for program compilation speed at the expense of software simplicity and development speed. On the other hand, the transition from a prototype compiler to production requires significant engineering work. To bridge this gap, we introduce the concept of sidekick compiler frameworks, an approach that uses multiple frameworks that interoperate with each other by leveraging textual interchange formats and declarative descriptions of abstractions. Each such compiler framework is specialized for specific use cases, such as performance or prototyping. Abstractions are by design shared across frameworks, simplifying the transition from prototyping to production. We demonstrate this idea with xDSL, a sidekick for MLIR focused on prototyping and teaching. xDSL interoperates with MLIR through a shared textual IR and the exchange of IRs through an IR Definition Language. The benefits of sidekick compiler frameworks are evaluated by showing on three use cases how xDSL impacts their development: teaching, DSL compilation, and rewrite system prototyping. We also investigate the trade-offs that xDSL offers, and demonstrate how we simplify the transition between frameworks using the IRDL dialect. With sidekick compilation, we envision a future in which engineers minimize the cost of development by choosing a framework built for their immediate needs, and later transitioning to production with minimal overhead.
cs.PL↗
Classical and Almost Sure Local Limit Theorems
We present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem, which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly concern sufficient conditions for the validity of the local limit theorem and its interesting variant forms: strong local limit theorem, strong local limit theorem with convergence in variation. Quite importantly are necessary conditions, and the results obtained are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary condition, and Mukhin's necessary and sufficient condition. Extremely useful and instructive are the counter-examples due to Azlarov and Gamkrelidze, as well as necessary and sufficient conditions obtained for a class of random variables, such as Mitalauskas' characterization of the local limit theorem in the strong form for random variables having stable limit distributions. The method of characteristic functions and the Bernoulli part extraction method, are presented and compared. A second part of the survey is devoted to the more recent study of the almost sure local limit theorem, instilled by Denker and Koch. The almost sure local limit theorems established already cover the i.i.d. case, the stable case, Markov chains, the model of the Dickman function. Our aim in writing this monograph was notably to bring to knowledge many interesting results obtained by the Lithuanian and Russian Schools of Probability during the sixties and after, and which are essentially written in Russian, and moreover often published in Journals of difficult access.
math.PR↗
Critical probabilistic characteristics of the Cram\'er model for primes and arithmetical properties
This work is a probabilistic study of the 'primes' of the Cram\'er model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\ \hbox{prime} \} \ge \frac{1}{\sqrt{2\pi e}\, }, \end{equation} and that for $b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2\pi B_n } } \int_{m_n-\sqrt{ 2bB_n\log n}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{-\frac{(t - m_n)^2}{ 2 B_n } }\, {\rm d}\pi(t), \end{equation} in which $m_n=\mathbb{E} S_n,B_n={\rm Var }\,S_n$, holds true for all $n\in \mathcal S$, $n\to \infty$. Further we prove that for any $0<\eta<1$, and all $n$ large enough and $ \zeta_0\le \zeta\le \exp\big\{ \frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n \xi_j$, \begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $\zeta$-quasiprime}\big\} \,\ge \, (1-\eta) \frac{ e^{-\gamma} }{ \log \zeta }, \end{eqnarray*} according to Pintz's terminology, where $c>0$ and $\gamma$ is Euler's constant. We also test which infinite sequences of primes are ultimately avoided by the 'primes' of the Cram\'er model, with probability 1. Moreover we show that the Cram\'er model has incidences on the Prime Number Theorem, since it predicts that the error term is sensitive to subsequences. We obtain sharp results on the length and the number of occurrences of intervals $I$ such as for some $z>0$, \begin{equation}\sup_{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z, \end{equation} which are tied with the spectrum of the Sturm-Liouville equation.
math.NT↗
On good universality and the Riemann hypothesis
We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelh{ö}f Hypothesis and the Riemann hypothesis. These ideas are then used to study the value distribution of Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. using Birkhoff's ergodic theorem and probability theory.
math.NT↗
On the Uniform Distribution (mod 1) of the Farey Sequence, quadratic Farey and Riemann sums with a remark on local integrals of $ζ(s)$
For $1$-periodic functions $f$ satisfying only a weak local regularity assumption of Dini's type at rational points of $]0,1[$, we study the Farey sums $$F_n(f)= \sum_{\frac{\k}ł\in \F_n} f\big(\frac{\k}ł\big),\qq F_{n,\s}(f)= \sum_{\frac{\k}ł\in \F_n} \frac{1}{\k^\sł^\s}f\big(\frac{\k}ł\big),\qq 1/2\le \s<1 , $$ where $\F_n$ is the Farey series of order $n\ge 1$. We obtain sharp estimates of $F_{n,\s}(f)$, for all $0< \s\le1$. We prove similar results for the corresponding Riemann quadratic sums $$ S_{n,\s}(f) \ =\ \sum_{1\le k\le \ell \le n}\frac{1}{(k\ell)^{\s }}\, f\big( \frac{k}{\ell}\big). $$ These sums are related to local integrals of the Riemann zeta-function over bounded intervals $I$, which are considered in the last part of the paper.
math.NT↗
An application of Brascamp-Lieb's inequality
We use Brascamp-Lieb's inequality to obtain new decoupling inequalities for general Gaussian vectors, and for stationary cyclic Gaussian processes. In the second case, we use a version by Bump and Diaconis of the strong Szego limit theorem. This extends results of Klein, Landau and Shucker.
math.PR↗
On the equation $ \boldsymbol{n_1n_2=n_3n_4}$ restricted to factor closed sets
We study the number of solutions $N(B,F)$ of the diophantine equation $n_1n_2=n_3n_4$, where $1\le n_1\le B$, $1\le n_3\le B$, $n_2, n_4\in F$ and $F\subset [1,B]$ is a factor closed set. We study more particularly the case when $F= \big\{m=p_1^{\e_1}\ldots p_k^{\e_k}, \e_j\in \{0,1\}, 1\le j\le k\big\}$, $p_1,\ldots,p_k$ being distinct prime numbers.
math.NT↗
On Nörlund summation and Ergodic Theory, with applications to power series of Hilbert contractions
We show that if ${\bf a}=(a_n)_{n\in \N}$ is a good weight for the dominated weighted ergodic theorem in $L^p$, $p>1$, then the Nörlund matrix $N_{\bf a}=\{a_{i-j}/A_i\}_{0\le j\le i}$, $A_i=\sum_{k=0}^i |a_k|$ is bounded on $\ell^p(\N)$. We study the regularity (convergence in norm, almost everywhere) of operators in ergodic theory: power series of Hilbert contractions, and power series $\sum_{n\in \N} a_nP_nf $ of $L^2$-contractions, and establish similar tight relations with the Nörlund operator associated to the modulus coefficient sequence $(|a_n|)_{n\in \N}$.
math.FA↗
Pointwise convergence of almost periodic Fourier series and associated series of dilates
Let $\mathcal S^2$ be the Stepanov space and let $ λ_n\uparrow\infty$. Let $(a_n)_{n\ge 1}$ be satisfying Wiener's condition $A:= \sum_{n\ge 1} \big(\sum_{k\, :\, n\le λ_k \le n+1}|a_k|\big)^2 <\infty$. We prove that $\big\| \sup_{N\ge 1} \big|\sum_{n=1}^Na_n{\rm e}^{iλ_n t}\big| \, \big\|_{\mathcal S^2}\le C\, A^{1/2} $ where $C>0$ denotes a universal constant. Moreover, the series $\sum_{n\ge 1} a_n{\rm e}^{itλ_n }$ converges for $λ$-a.e. $t\in \mathbb R$. This contains as a special case Hedenmalm and Saksman result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let $1\le p,q\le 2$ be such that $1/p+1/q=3/2$. Then for any sequence $(α_n)_{n\ge 1}$ and $(β_n)_{n\ge 1}$ of complex numbers such that $K:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le λ_k< n+1}|α_k|\,\big)^p <\infty$ and $L:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le μ_k< n+1} |β_k|\,\big)^q <\infty$, we have $$ \Big\|\sup_{N\ge 1} \big|\sum_{n=1}^N α_n D(λ_n t)\big|\, \Big\|_{\mathcal S^2} \le C\, K^{1/p}\, L^{1/q } $$ where $D(t)= \sum_{n\ge 1}β_n {\rm e}^{iμ_n t}$ is defined in $\mathcal S^2$. Moreover, the series $\sum_{n\ge 1} α_n D(λ_nt)$ converges in $\mathcal S^2$ and for $λ$-a.e. $t\in \mathbb R$. We further show that if $\{λ_k, k\ge 1\}$ satisfies the following condition $$\sum_{ k\not=\ell\,,\, k'\not=\ell'\atop (k,\ell)\neq(k',\ell')}\big(1-|(λ_k-λ_\ell)-(λ_{k'}-λ_{\ell'}) |\big)_+^2 \, <\infty,$$ then the series $\sum_{k} a_k {\rm e}^{iλ_kt}$ converges on a set of positive Lebesgue measure, only if the series $\sum_{k=1}^\infty |a_k|^2$ converges. The above condition is in particular fulfilled when $\{λ_k, k\ge 1\}$ is a Sidon sequence.
math.CA↗
Criteria of Divergence Almost Everywhere in Ergodic Theory
In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain's entropy criteria and Kakutani-Rochlin lemma, most classical device for these questions in ergodic theory. First, we state a $L^1$-version of the continuity principle and give an example of its usefulness by applying it to some famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in $L^p$, $2\le p\le \infty$ and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on $L^2$. We further study the corresponding criterion in $L^p$, $1<p<2$ using properties of $p$-stable processes. Finally we consider Kakutani-Rochlin's lemma, one of the most frequently used tool in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages.
math.DS↗
On the $Φ$-variation of stochastic processes with exponential moments
We obtain sharp sufficient conditions for exponentially integrable stochastic processes $X=\{X(t)\!\!: t\in [0,1]\}$, to have sample paths with bounded $Φ$-variation. When $X$ is moreover Gaussian, we also provide a bound of the expectation of the associated $Φ$-variation norm of $X$. For an Hermite process $X$ of order $m\in \N$ and of Hurst index $H\in (1/2,1)$, we show that $X$ is of bounded $Φ$-variation where $Φ(x)=x^{1/H}(\log(\log 1/x))^{-m/(2H)}$, and that this $Φ$ is optimal. This shows that in terms of $Φ$-variation, the Rosenblatt process (corresponding to $m=2$) has more rough sample paths than the fractional Brownian motion (corresponding to $m=1$).
math.PR↗
Local Limit Theorems in some Random models from Number Theory
We study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we consider arise from standard random models used in arithmetical number theory. We next use the characteristic function method to prove new local limit theorems for weighted sums of Bernoulli variables. Further, we give an application of the almost sure local limit theorem to a representation problem in additive number theory due to Burr, using an appropriate random model. We also give a simple example showing that the local limit theorem, in its standard form, fails to be sharp enough for estimating the probability $P\{S_n\in E\}$ for infinite sets of integers $E$, already in the simple case where $S_n$ is a sum of $n$ independent standard Bernoulli random variables and $E$ an arithmetic progression.
math.PR↗
Cauchy Means of Dirichlet polynomials
We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q} \frac{\dd t}{π( t^2+1)}.$$ These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\ge 1$, $\s\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.
math.NT↗
Ergodic theorems with arithmetical weights
We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},ν,τ)$ and any $f\in L^p(ν)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(τ^k x)$$ exists $ν$-almost everywhere. We also obtain similar results for other arithmetical functions, like $θ(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the circle method based on the shift model.
math.DS↗
An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function
Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m \sin {2πm x}$ where $\sum_{m=1}^\infty a_{m }^2d(m) <\infty$ and $d(m)=\sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $K\subset \N$ finite, $ \|\sum_{k\in K} c_k f_k \|_2^2 \le ( \sum_{m=1}^\infty a_{m }^2 d(m) ) \sum_{k\in K } c_{k}^2d(k^2)$. If $f^s (x)= \sum_{j=1}^\infty \frac{\sin 2πjx}{j^s}$, $s>1/2$, by only using elementary Dirichlet convolution calculus, we show that for $0< \e\le 2s-1$, $ζ(2s)^{-1} \|\sum_{k\in K} c_k f^s_k \|_2^2 \le \frac{1+\e}{\e } (\sum_{k \in K} |c_k|^2 \s_{ 1+\e-2s}(k) )$, where $\s_h(n)=\sum_{d|n}d^h$. From this we deduce that if $f\in {\rm BV}(\T)$, $\langle f,1\rangle=0$ and $$\sum_{k} c_k^2\frac{(\log\log k)^4}{(\log\log \log k)^2} <\infty,$$ then the series $ \sum_{k } c_kf_k$ converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc (\cite{ABS}, th.3) ($n_k=k$), where it was assumed that $ \sum_{k} c_k^2 \, (\log\log k)^\g $ converges for some $\g>4$. We further show that the same conclusion holds under the arithmetical condition $$\sum_{ k } c_k^2 (\log\log k)^{2 + b} \s_{ -1+\frac{1}{(\log\log k)^{ b/3}} }(k) <\infty,$$ for some $b>0$, or if $ \sum_{ k} c_k^2 d(k^2) (\log\log k)^2 <\infty$. We also derive from a recent result of Hilberdink an $Ø$-result for the Riemann Zeta function involving factor closed sets. We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series $\sum_{k=0}^\infty c_k f_k $.
math.NT↗
Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery
We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.
math.PR↗