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We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, the endpoint…

We consider data transmission across discrete memoryless channels (DMCs) using variable-length codes with feedback. We consider the family of such codes whose rates are $\rho_N$ below the channel capacity $C$, where $\rho_N$ is a positive…

We present an upper bound on the error probability achievable using variable-length stop feedback codes, for a fixed size of the information payload and a given constraint on the maximum latency and the average service time. Differently…

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…

For a $\psi$-mixing process $\xi_0,\xi_1,\xi_2,...$ we consider the number $\mathcal{N}_N$ of multiple returns $\{\xi_{q_{i,N}(n)}\in\Gamma_N,\, i=1,...,\ell\}$ to a set $\Gamma_N$ for $n$ until either a fixed number $N$ or until the moment…

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.

In 2013, Bollob\'as, Mitsche, and Pralat at gave upper and lower bounds for the likely metric dimension of random Erd\H{o}s-R\'enyi graphs $G(n,p)$ for a large range of expected degrees $d=pn$. However, their results only apply when $d \ge…

For any discrete probability distributions with bounded entropy, we can generate exactly a random variate using only a finite expected number of perfect coin flips. A perfect coin flip is the outcome of an unbiased Bernoulli random…

Hilberg's conjecture about natural language states that the mutual information between two adjacent long blocks of text grows like a power of the block length. The exponent in this statement can be upper bounded using the pointwise mutual…

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known,…

We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by…

This paper introduces a simple Markov process inspired by the problem of quasicrystal growth. It acts over two-letter words by randomly performing \emph{flips}, a local transformation which exchanges two consecutive different letters. More…

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We estimate simultaneously the drift and shift parameters. On the one hand, we establish a large deviation…

Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences…

In a recent work it was suggested that the number of limit cycles in a piecewise-linear system could be closely related to the number of zones, that is the number of parts of the phase plane where the system is linear. In this note we…

The problem of the order of the fluctuation of the Longest Common Subsequence (LCS) of two independent sequences has been open for decades. There exist contradicting conjectures on the topic, due to Chvatal - Sankoff in 1975 and Waterman in…

We derive large-sample and other limiting distributions of the ``frequency of frequencies'' vector, ${\bf M_n}$, together with the number of species, $K_n$, in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling…

We consider random walks on non-amenable Baumslag-Solitar groups BS(p,q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it…

On a finite graph, there is a natural family of Boltzmann probability measures on cycle-rooted spanning forests, parametrized by weights on cycles. For a certain subclass of those weights, we construct Gibbs measures in infinite volume, as…

A significant obstacle in the development of robust machine learning models is covariate shift, a form of distribution shift that occurs when the input distributions of the training and test sets differ while the conditional label…