Related papers: Poisson-Dirichlet approximation for counting integ…

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…

Suppose that $\ff \in \reals^{n}$ is a vector of $n$ error-contaminated measurements of $n$ smooth values measured at distinct and strictly ascending abscissae. The following projective technique is proposed for obtaining a vector of smooth…

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some…

An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the…

In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a…

Let $p_1 \ge p_2 \ge \dots$ be the prime factors of a random integer chosen uniformly from $1$ to $n$, and let $$ \frac{\log p_1}{\log n}, \frac{\log p_2}{\log n}, \dots $$ be the sequence of scaled log factors. Billingsley's Theorem…

In this paper, we develop simple, yet efficient, procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process. We compare the efficiency of the new approximations to the…

We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We…

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general…

We consider a random interval splitting process, in which the splitting rule depends on the empirical distribution of interval lengths. We show that this empirical distribution converges to a limit almost surely as the number of intervals…

This paper focuses on the problem of finding a distribution for an associated entropic vector in the entropy space nearest to a given, possibly non-entropic, target vector for random variables with a constraint on alphabet size. We show the…

If $\alpha$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the Dirichlet random probability $P_t\sim\mathcal{D}(t\alpha) ;$ it is such that for any measurable partition $(A_0,\ldots,A_k)$ of $\mathbb{R}^d$ then…

Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian…

We revisit the problem of finding small $\epsilon$-separation keys introduced by Motwani and Xu (2008). In this problem, the input is $m$-dimensional tuples $x_1,x_2,\ldots,x_n $. The goal is to find a small subset of coordinates that…

This paper considers a probabilistic-analytical approach to determining asymptotics of prime objects on the initial interval of the natural series. The author proposes a new method based on the construction of a probability space. An…

In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several…

We consider the problem of counting $k$-tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than $x$. We first establish the conditions to guarantee the existence of such tuples, and…

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$,…

We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(a,0)$. Precisely, let $s$ be a proper random…

We show that for all large enough $x$ the interval $[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the…