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An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form…
Let $\{X_n\}_{n\ge0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(\alpha):=n^{-1}\sum_{k=1}^nF(\alpha,X_{k-1},X_k)$, $\alpha\in\mathcal{A}\subset \mathbb {R}$. Consider an $M$ estimator…
We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain…
Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the…
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular,…
Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next,…
We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le1/2(1+\sqrt{5})$, then $L_n$…
A celebrated result of Otter says the number of distinct unlabelled spanning trees in $K_n$ is $\alpha^n$ up to subexponential factors for an absolute constant $\alpha>0$. In this note, we prove that for every $0<\varepsilon<\alpha$, there…
We consider $m$ spinless Fermions in $l > m$ degenerate single-particle levels interacting via a $k$-body random interaction with Gaussian probability distribution and $k <= m$ in the limit $l$ to infinity (the embedded $k$-body random…
The randomized $k$-number partitioning problem is the task to distribute $N$ i.i.d. random variables into $k$ groups in such a way that the sums of the variables in each group are as similar as possible. The restricted $k$-partitioning…
Statistical inference on biodiversity has a rich history going back to RA Fisher. An influential ecological theory suggests the existence of a fundamental biodiversity number, denoted $\alpha$, which coincides with the precision parameter…
In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_\alpha(t)$, $N_\beta(t)$, $t>0$, we show that $N_\alpha(N_\beta(t))…
Non-Maxwellian distributions and their origins in space plasma have attracted significant attention due to their prevalence and impact on various astrophysical and space-related phenomena. This paper presents a theoretical study of the…
We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We…
The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$s$ expansion with two redundant digits form a…
The recently introduced two-parameter Poisson-Dirichlet diffusion extends the infinitely-many-neutral-alleles model, related to Kingman's distribution and to Fleming-Viot processes. The role of the additional parameter has been shown to…
Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $\Omega_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on…
Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(\alpha x_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\alpha$…
We develop a model based on a generalised Poisson-Dirichlet distribution for the analysis of genetic diversity, and illustrate its use on microsatellite data for the genus Dasyurus (the quoll, a marsupial carnivore listed as near-threatened…
In this paper, we study the effect of sparsity on the appearance of outliers in the semi-circular law. Let $(W_n)_{n=1}^\infty$ be a sequence of random symmetric matrices such that each $W_n$ is $n\times n$ with i.i.d entries above and on…