Related papers: Maximum of Exponential Random Variables, Hurwitz's…
This paper develops upper and lower bounds for the probability of Boolean expressions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. Our technique generalizes and extends the…
The partition function of the random energy model at inverse temperature $\beta$ is a sum of random exponentials $Z_N(\beta)=\sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables…
This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let…
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 was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two…
Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any $n\neq 7$. The basis for this is an inequality for the partition function which…
Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting…
Let $P_N$ be a uniform random $N\times N$ permutation matrix and let $\chi_N(z)=\det(zI_N- P_N)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically, \[…
We introduce a new framework term coding for extremal problems in discrete mathematics and information flow, where one chooses interpretations of function symbols so as to maximise the number of satisfying assignments of a finite system of…
The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's…
The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…
We study the asymptotic behavior of the maximal multiplicity $M_n=M_n(\sigma)$ of the blocks in a set partition of $[n]=\{1,2,...,n\}$, assuming that $\sigma$ is chosen uniformly at random from the set of all such partitions. Let $W=W(n)$…
We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…
Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using…
We obtain the distribution of the maximal average in a sequence of independent identically distributed exponential random variables. Surprisingly enough, it turns out that the inverse distribution admits a simple closed form. An application…
An algorithm is presented which, with optimal efficiency, solves the problem of uniform random generation of distribution functions for an n-valued random variable.
We give new integral and series representations of the Hurwitz zeta function. We also provide a closed-form expression of the coefficients of the Laurent expansion of the Hurwitz-zeta function about any point in the complex plane.
The sequential sampling of populations with unequal probabilities and with replacement in a closed population is a recurrent problem in ecology and evolution. Many of these questions can be reformulated as urn problems, often as special…
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…
We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an…