Related papers: Renyi's Parking Problem Revisited
A parking function is a function $\pi:[n]\to [n]$ whose $i$th-smallest output is at most $i,$ corresponding to a parking procedure for $n$ cars on a one-way street. We refine this concept by introducing preference-restricted parking…
Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for a uniformly random permutation ${\boldsymbol{\pi}}$ of…
In this article a multilayer parking system with screening of size n=3 is studied with a focus on the time-dependent particle density. We prove that the asymptotic limit of the particle density increases from an average density of 1/3 on…
Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of…
R\'enyi entropy is an important measure in the context of information theory as a generalization of Shannon entropy. This information measure was often used for uncertainty quantification of dynamical behaviour of stochastic processes. In…
The \emph{Shi arrangement} is the set of all hyperplanes in $\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \le j < k \le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this…
First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to…
In this article, we establish new results on the probabilistic parking model (introduced by Durm\'ic, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in…
For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of $Z_n$,…
We introduce the Random Subsequence Model, a spin glass model on pairs of random strings $(X,Y) \in \{0,1\}^N \times \{0,1\}^M$ whose partition function counts subsequence embeddings of $Y$ into $X$. We study two variants: the null model,…
As urban populations continue to grow, cities face numerous challenges in managing parking and determining occupancy. This issue is particularly pronounced in university campuses, where students need to find vacant parking spots quickly and…
We extend the Rate-Distortion-Perception (RDP) framework to the R\'enyi information-theoretic regime, utilizing Sibson's $\alpha$-mutual information to characterize the fundamental limits under distortion and perception constraints. For…
Neural Processes (NPs) are deep probabilistic models that represent stochastic processes by conditioning their prior distributions on a set of context points. Despite their advantages in uncertainty estimation for complex distributions, NPs…
An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure,…
We study the variational inference problem of minimizing a regularized R\'enyi divergence over an exponential family. We propose to solve this problem with a Bregman proximal gradient algorithm. We propose a sampling-based algorithm to…
A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum $\rho$-th moment of the number of performed…
This paper proposes a totally asymmetric simple exclusion process on a traveling lane, which is equipped with a queueing system and functions of site assignments along the parking lane. In the proposed system, new particles arrive at the…
We shift the perspective on the interval fragmentation problem from division points to division spacings. This leads to a proof that is both simpler and stronger, establishing limiting distributions for partition points and spacings and,…
Confidence interval procedures used in low dimensional settings are often inappropriate for high dimensional applications. When a large number of parameters are estimated, marginal confidence intervals associated with the most significant…
A special type of binomial splitting process is studied. Such a process can be used to model a high-dimensional corner parking problem, as well as the depth of random PATRICIA tries (a special class of digital tree data structures). The…