Related papers: On the recent-$k$-record of discrete random variab…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in R^n$ the random variables $\sum a_iX_i$ and $\gamma(a) Y$ are identically distributed, where $\gamma:R^n\to…

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of $k$-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of $k$-regular graphs: given…

We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It…

A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the…

We consider univariate regression estimation from an individual (non-random) sequence $(x_1,y_1),(x_2,y_2), ... \in \real \times \real$, which is stable in the sense that for each interval $A \subseteq \real$, (i) the limiting relative…

We determine the probability distribution for relative projective objects in an exceptional sequence of type $A_n$ of any length. We show that these events (the $j$-th object in an exceptional sequence of length $k\le n$ being relatively…

In [8] the author of this paper continued the research on the complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N)$ recently introduced and studied in [24]. Here we extend our results by considering $X_l(m,N)$…

Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erd\H{o}s-R\'enyi-Gilbert…

Suppose $k$ balls are dropped into $n$ boxes independently with uniform probability, where $n, k$ are large with ratio approximately equal to some positive real $\lambda$. The maximum box count has a counterintuitive behavior: first of all,…

Consider a sequence of Poisson random connection models (X_n,lambda_n,g_n) on R^d, where lambda_n / n^d \to lambda > 0 and g_n(x) = g(nx) for some non-increasing, integrable connection function g. Let I_n(g) be the number of isolated…

In this paper, we study the hypothesis testing problem of, among $n$ random variables, determining $k$ random variables which have different probability distributions from the rest $(n-k)$ random variables. Instead of using separate…

Let $(\xi_k, \eta_k)_{k\geq 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:=\xi_1+\ldots+\xi_{k-1}+\eta_k$for $k\in\mathbb{N}$. We call the random sequence {T_k, k=1,2...} a…

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…

A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the…

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a…

Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $ such that $r \le n / 2$, its vertices are all the subsets of $[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the corresponding subsets…

We show how to extend several basic concentration inequalities for simple random tensors $X = x_1 \otimes \cdots \otimes x_d$ where all $x_k$ are independent random vectors in $\mathbb{R}^n$ with independent coefficients. The new results…

The validity of the strong law of large numbers for multiple sums $S_n$ of independent identically distributed random variables $Z_k$, $k\leq n$, with $r$-dimensional indices is equivalent to the integrability of $|Z|(\log^+|Z|)^{r-1}$,…

Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we…