Related papers: A Probabilistic Proof of a Wallis-type Formula for…
A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict…
A simple application of the semipositivity.
We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments.
This paper presents the asymptotic distributions of a general likelihood-based test statistic, derived using results of Wilks and Wald. The general form of the test statistic incorporates the test statistics and associated asymptotic…
In a previous work (arXiv:2505.05574), a summation formula for harmonic Maass forms of polynomial growth was established. In this note, we use the theory of $L$-series of harmonic Maass forms to state and prove a summation formula for such…
We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability…
We show that there are four possibilities for the product of all elements in the multiplicative group of a quotient of the ring of integers in a number field, and give precise conditions for each of the possibilities to occur. This…
In this paper, we have proposed a new method for solving the Gaussian integral. Introducing a parameter that depends on a $n$ index, we have found a general solution for this type of integral inspired by Taylor series of a simple function.…
In the paper, we establish an inequality involving the gamma and digamma functions and use it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.
We characterize the bialgebraic varieties of the $\Gamma$ function, that is, if $V,W\subseteq\mathbb{C}^n$ are irreducible affine algebraic variety which satisfy $\dim V =\dim W$ and $\Gamma(V)\subseteq W$, then the equations defining $V$…
The notion of belief likelihood function of repeated trials is introduced, whenever the uncertainty for individual trials is encoded by a belief measure (a finite random set). This generalises the traditional likelihood function, and…
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original…
We derive the probability distribution of product of two independent random variables, each distributed according the one-dimensional stable law. We represent the density by its power series and its asymptotic expansions. As Fox's…
We show that if we require positive definite probabilities, then frequently cited results of Prato, of Lenzi and Hilhorst on the nonextensive equilibrium statistical mechanics of gases are valid only for a limited range of the Tsallis…
Dependency functions of dependent variables are relevant for i) performing uncertainty quantification and sensitivity analysis in presence of dependent variables and/or correlated variables, and ii) simulating random dependent variables. In…
It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…
Bivariate partial-sums discrete probability distributions are defined. The question of the existence of a limit distribution for iterated partial summations is solved for finite-support bivariate distributions which satisfy conditions under…
A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions…
While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a…
Analogies between codes and lattices have been extensively studied for the last decades, in this dictionary, the MacWilliams identity is the finite analog of the Jacobi-Poisson formula of the Theta function. Motivated by the random theory…