arXiv subjects
Publications and source records attributed to Aristides V. Doumas.
A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are illustrated. Furthermore, several asymptotic results are discussed and a new expression related to the Riemann hypothesis is presented.
In this paper we study two variants of the generalized coupon collector's problem, where our collector receives at each run d distinct coupons and keeps all the new observed coupons (Problem I), while he chooses the least--collected coupon at each run (Problem II). In both cases we derive explicit formulae for the average of the random variable denoting the number of trials for a complete set of N different types of coupons, which are uniformly distributed. In both cases we present the asymptotic expansion up to the fourth term including the corresponding error term. Then, for both problems we derive the full asymptotic expansion as N\rightarrow \infty. We further obtain the leading-order behaviour of the variance, showing that in both problems \mathrm{Var}\sim \frac{π^2}{6}\frac{N^2}{d^2}, and we establish a rate of convergence to the limiting law. Our analysis is based on the Nørlund--Rice integral method applied to an alternating binomial sum and classical tools from asymptotic analysis. The leading asymptotic term for Problem II was obtained by W. Xu and A. K. Tang [\textit{J. Appl. Probab.} \textbf{48} (2011), 1081--1094]. Finally, for both problems, we derive the limiting distribution under the appropriate normalization. As expected, the limit is standard Gumbel; however, the normalization differs between Problems I and II. As an application, we show that Problem~I describes exactly the sequencing-coverage process in combinatorial motif-based DNA data storage, and our expansions yield closed-form coverage estimates for that setting.
In the siblings (or brotherhood) variant of the coupon collector's problem, a main collector draws coupons until her own album is complete and passes every duplicate down a chain of siblings; the $j$th collector is then left with $U_j^N$ empty places, $j\ge 2$. It has been conjectured [stated as an open problem in the work that introduced the model] that, for every fixed number of coupon types $N$ and every $j\ge 2$, the expected deficit $\E[U_j^N]$ is maximized by the equiprobable coupon distribution. We prove this in a sharp, finite-$N$ form: $\E[U_j^N]$ is strictly larger at the uniform vector than at any other probability vector, and indeed strictly increases along every ray running from an arbitrary distribution toward the uniform one. The proof is exact and elementary in its ingredients. An inclusion--exclusion step turns the governing Poissonized integral into a one-dimensional integral with a separable integrand; a single integration by parts then rewrites the radial derivative of $\E[U_j^N]$ as a positively weighted covariance of an increasing function, whose sign is settled by Chebyshev's correlation inequality. We show that $\E[U_j^N]$ is \emph{not} Schur-concave, so that no majorization or pairwise-smoothing argument can yield the result, and we explain why the recent variance-extremality method of Long~[Long, arXiv:2604.25108, 2026] does not transfer. As by-products we obtain a finite closed form for $\E[U_j^N]$ over subsets of the coupon set and the exact Hessian of $\E[U_j^N]$ at the uniform vector. The argument extends without change to all real $j>1$.
This paper develops Kolmogorov-type maximal inequalities for sums of Negative Binomial random variables under both independence and dependence structures. For independent heterogeneous Negative Binomial variables we derive sharp Markov-type deviation inequalities and Kolmogorov-type bounds expressed in terms of Tweedie dispersion parameters, providing explicit control limits for NB2 generalized linear model monitoring. For dependent count data arising through a shared Gamma mixing variable, we establish a \emph{sub-exponential Bernstein-type refinement} that exploits the Poisson-Gamma hierarchical structure to yield exponentially decaying tail probabilities -- this refinement is new in the literature. Through moment-matched Monte Carlo experiments ($n=20$, 2{,}000 replications), we document a 55\% reduction in mean maximum deviation under appropriate dependence structures, a stabilization effect we explain analytically. A concrete epidemiological application with NB2 parameters calibrated from COVID-19 surveillance data demonstrates practical utility. These results materially advance the applicability of classical maximal inequalities to overdispersed and dependent count data prevalent in public health, insurance, and ecological modeling.
In this letter, we prove an inequality involving alternating binomial logarithmic sums by exploiting the variance of the logarithm of the maximum of independent and identically distributed exponential random variables. This inequality was introduced in our recent work [On the minimum of independent collecting processes via the Stirling numbers of the second kind, Statist. Probab. Lett., 185 (2022)].
The double Dixie cup problem of D.J. Newman and L. Shepp is a well-known variant of the coupon collector problem, where the object of study is the number of coupons that a collector has to buy in order to complete m sets of all N existing different coupons. In this paper we consider the case where the coupons distribution is a mixture of two different distributions, where the coupons from the first distribution are far rarer than the ones coming from the second. We apply a Poissonization technique, as well as well known results and techniques from our previous work, to derive the asymptotics (leading term) of the expectation of the above random variable as N goes to infinity for large classes of distributions. As it turns out, both distributions contribute to this result. The leading asymptotics of the rising moments of the aforementioned random variable are also discussed. We conclude by generalizing the problem to the case where the family of coupons is a mixture of j subfamilies.
We consider the combinatorial problem where $p$ players aim to a complete set of $N$ different types of items (species) which are uniformly distributed. Let the random variables $T_{N(i)},\,\,i=1,2,\cdots,p$ denoting the number of trials needed until all $N$ types are detected (at least once), respectively for each player. This paper studies the impact of the number $p$ in the asymptotics of the expectation, the second moment, and the variance of the random variable \begin{equation*} M_{N(p)}: = \bigwedge_{i=1}^p T_{N(i)},\,\,\,\,\,\,N\rightarrow \infty. \end{equation*} The main ingredient in the expression of these quantittes are sums involving the Stirling numbers of the second kind; for which the asymptotics are explored. At the end of the paper we conjecture on a remarkable \textit{combinatorial identity}, regarding alternating binomial sums. These sums have been studied (mainly) by P. Flajolet due to their applications to digital search trees and quadtrees.
Consider a broadcast channel with $n$ users, where different users receive different messages, and suppose that each user has to receive $m$ packets. A quantity of interest here, introduced by Sharif and Hassibi (2006-7) \cite{Sh}, \cite{S-H}, is the (\emph{packet}) \emph{delay} $D_{m,n}$, namely the number of channel uses required to guarantee that all users will receive $m$ packets. For the case of a \emph{homogeneous} network, where in each channel use the transmitter chooses a user at random, i.e. with probability $1/n$, and sends him$/$her a packet, the same quantity $D_{m,n}$ had already appeared in the \emph{coupon collector} context, in the works of Newman and Shepp (1960) \cite{N-S} and of Erdős and Rényi (1961) \cite{E-R}. A problem of particular interest in wireless communications, related to the delay $D_{m,n}$, is to determine its behavior as $n$ and $m$ grow large. Regarding this problem, Sharif and Hassibi \cite{Sh}, \cite{S-H} managed to calculated the asymptotics of the mean value $\mathbb{E}[D_{m,n}]$, as $n \to \infty$, for the cases (a) $m = \ln n$ and (b) $m = (\ln n)^ρ$, $ρ> 1$. It is remarkable that Erdős and Rényi \cite{E-R} had, also, raised the question of the determination of the asymptotic profile of $D_{m,n}$ for large $m$ and $n$ (in 1961). And in the 1970's the limiting distribution of $D_{m,n}$ for large $m$ and $n$ was determined (in great generality) by Ivchenko \cite{I1} and \cite{I2}. In this article we determine the asymptotics of the moments of $D_{m,n}$ for large $m$ and $n$. We also derive its limiting distribution in the "supercritical case" where $m$ grows faster than $\ln n$ and in the "critical case" $m \sim β\ln n$, by an approach which is different from the one used by Ivchenko.
A popular variant of the collector's problem is the following: Assume there are $N$ different types of coupons with equal occurring probabilities. There is one main collector who collects coupons. When she gets a "double," she gives it to her younger brother. Hence, when the main collector completes her collection, her brother's album will still have, say, $U_N$ empty spaces. We show that, as $N \to \infty$, the limiting distribution of $U_N/\ln N$ is exponential with parameter $1$.
We consider the following variant of the classic collector's problem: The family of coupon probabilities is the mixing of two subfamilies one of which is the \textit{uniform} family, while the other belongs to the well known \textit{Zipf family}. We obtain asymptotics for the expectation, the second rising moment, and the variance of the random variable $T_N$, namely the number of trials needed for all the $N$ types of coupons to be collected (at least once, with replacement) as $N \rightarrow \infty$. It is interesting that the effect of the uniform subcollection on the asymptotics of the expectation of $T_N$ (at least up to the sixth term) appears only in the leading factor of the expectation of $T_N$. The limiting distribution of $T_N$ is derived as well. These results answer a question placed in a recent work of ours [\textit{Electron. J. Probab.} \textbf{18} (2012) 1--15].
A collector samples coupons with replacement from a pool containing $g$ \textit{uniform} groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur. For each $j = 1, \dots, g$ let $T_j$ be the number of trials needed to detect Group $j$, namely to collect all $M_j$ coupons belonging to it at least once. We derive an explicit formula for the probability that the $l$-th group is the first one to be detected (symbolically, $P\{T_l = \bigwedge_{j=1}^g T_j\}$). We also compute the asymptotics of this probability in the case $g=2$ as the number of coupons grows to infinity in a certain manner. Then, in the case of two groups we focus on $T := T_1 \vee T_2$, i.e. the number of trials needed to collect all coupons of the pool (at least once). We determine the asymptotics of $E[T]$ and $V[T]$, as well as the limiting distribution of $T$ (appropriately normalized) as the number of coupons becomes very large.
The "double Dixie cup problem" of D.J. Newman and L. Shepp (1960) is a well-known variant of the coupon collector's problem, where the object of study is the number $T_{m}(N)$ of coupons that a collector has to buy in order to complete $m$ sets of all $N$ existing different coupons. More precisely, the problem is to determine the asymptotics of the expectation (and the variance) of $T_{m}(N)$, as well as its limit distribution, as the number $N$ of different coupons becomes arbitrarily large. The classical case of the problem, namely the case of equal coupon probabilities, is here extended to the general case, where the probabilities of the selected coupons are unequal. In the beginning of the article we give a brief review of the formulas for the moments and the moment generating function of the random variable $T_{m}(N)$. Then, we develop techniques of computing the asymptotics of the first and the second moment of $T_{m}(N)$ (our techniques apply to the higher moments of $T_{m}(N)$ as well). From these asymptotic formulas we obtain the leading behavior of the variance $V[\,T_{m}(N)\,]$ as $N \to \infty$. Finally, based on the asymptotics of $E[\,T_{m}(N)\,]$ and $V[\,T_{m}(N)\,]$ we obtain the limit distribution of the random variable $T_{m}(N)$ for large classes of coupon probabilities. As it turns out, in many cases, albeit not always, $T_{m}(N)$ (appropriately normalized) converges in distribution to a Gumbel random variable. Our results on the limit distribution of $T_{m}(N)$ generalize a well-known result of P. Erdős and A. Rényi (1961) regarding the limit distribution of $T_{m}(N)$ for the case of equal coupon probabilities.
A collector wishes to collect $m$ complete sets of $N$ distinct coupons. The draws from the population are considered to be independent and identical distributed with replacement, and the probability that a type-$j$ coupon is drawn is noted as $p_{j}$. Let $T_{m}(N)$ the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of $T_{m}(N)$ (leading term), as well as its limit distribution as $N\rightarrow \infty$, when \begin{equation*} p_{j}=\frac{a_{j}}{\sum_{j=2}^{N+1} a_{j}}, \,\,\,\text{where}\,\,\, a_{j}=\left(\ln j\right)^{-p}, \,\,p>0. \end{equation*} These "log-Zipf" classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. Therefore, we enlarge the classes for which the collector's problem is solved (moments, variance, distribution).
We propose a stochastic version of the Collatz $3x + 1$ Problem.
The following variant of the collector's problem has attracted considerable attention relatively recently (see, e.g., N. Pintacuda 1980, D. Foata H. Guo-Niu and B. Lass 2001, D. Foata and D. Zeilberger 2003, I. Adler, S. Oren and S. Ross 2003, and S. Ross 2010): There is one main collector who collects coupons. Assume there are $N$ different types of coupons with, in general, unequal occurring probabilities. When the main collector gets a "double", she gives it to her older brother; when this brother gets a "double", he gives it to the next brother, and so on. Hence, when the main collector completes her collection, the album of the $j$-th sibling, $j = 2, 3, \dots$, will still have $U_j^N$ empty spaces. In this article we develop techniques of computing asymptotics of the average $E[U_j^N]$ of $U_j^N$ as $N \rightarrow \infty,$ for a large class of families of coupon probabilities. We also give various illustrative examples.