Related papers: A Central Limit Theorem for intransitive dice

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian…

For a given Dirichlet character $\chi (n) = e^{i \theta_n}$, we prove central limit theorems for the series $\sum_{p'} \cos \theta_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$…

We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular…

The central limit theorem provides the theoretical foundation for the universality of the normal distribution: under broad conditions, the asymptotic distribution of a sum of independent random variables approaches a Gaussian. Yet, physical…

The prediction of the final state probabilities of a general cuboid randomly thrown onto a surface is a problem that naturally arises in the minds of men and women familiar with regular cubic dice and the basic concepts of probability.…

We give a central limit theorem, which has applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced generalized Polya urns.

In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each…

Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove…

We provide a systematic approach to stable central limit theorems for d-dimensional martingale difference arrays and martingale difference sequences. The conditions imposed are straightforward extensions of the univariate case.

This recreational mathematics article shows that the game of Snakes and Ladders is intransitive: square 69 has a winning edge over 79, which in turn beats 73, which beats 69. Analysis of the game is a nice illustration of Markov chains,…

Penney's Ante exhibits non-transitivity when two target strings race to appear in a shared stream of coin tosses. We study instead independent string races, where each player observes their own independent and identically distributed…

Suppose X is a random vector, that is distributed uniformly in some n-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector u, such that the distribution of the real random…

A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…

We get central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on distributions of the weights.

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…

We consider a simple dice game, which leads to an intriguing study of multinomial walks, with surprising and seemingly paradoxical properties. The winning and losing probabilities of a general version of the game are investigated via…

Let $f$ be a $C^{2+\epsilon}$ expanding map of the circle and $v$ be a $C^{1+\epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = \alpha (f(x)) - Df(x) \alpha (x)$ which has a unique bounded solution…

For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which…

We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number…

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each,…