arXiv subjects

Italo Simonelli

Publications and source records attributed to Italo Simonelli.

3 recordsLinked to original sources

arXiv2023

A central limit theorem for a card shuffling problem

Given a positive integer $n$, consider a random permutation $\tau$ of the set $\{1,2,\ldots, n\}$. In $\tau$, we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each block in $\tau$ is merged, and after all the merges, the elements of this new set are relabeled from $1$ to the current number of elements. We continue to randomly permute and merge this new set until only one integer is left. In this paper, we investigate the asymptotic behavior of $X_n$, the number of permutations needed for this process to end. In particular, we find an explicit asymptotic expression for each of $\mathbf{E}[X_n]$ and $\mathbf{Var} [X_n]$ as well as for every higher central moment, and show that $X_n$ satisfies a central limit theorem.

math.PR↗

arXiv2021

Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.

math.CO↗

arXiv2018

Local Limit Theorems for Poisson's Binomial in the Case of Infinite Expectation

Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $λ_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $λ= \lim\limits_{n \to \infty} λ_n,$ and $m_n = \max\limits_{1 \leq i \leq n} b(i;n)$. We derive asymptotic results for $P(V_{n}=k)$ that hold without assuming that $λ< +\infty$ or $m_n \to 0$. Also, we do not assume $k$ to be fixed, but instead, our results hold uniformly for all $k$ which satisfy particular growth conditions with respect to $n$. These results extend known Poisson local limit theorems to the case when $λ= +\infty$. While our results apply to triangular arrays, without the assumption that $m_n \to 0$ they continue to hold for sums of Bernoulli random variables. In this setting, our growth conditions cover a range of values for $k$ not centered at $λ_n$, thus complementing known local limit theorems based on approximation by the normal distribution. In addition, we show that our local limit theorems apply to a scheme of dependent random variables introduced in the work of Sevast'yanov.

math.PR↗