Related papers: The rencontre problem

Let $\{B_k\}_{k=1}^\infty, \{X_k\}_{k=1}^\infty$ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty$ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text{dist}}{=}\text{Ber}(p_k)$, with…

We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $\alpha$…

Let m_j denote the jth moment of a random variable X. One of the results is the inequality m_3 \leq c m_4^{3/4} with the best possible factor c over all random variables X with a non-positive mean.

The purpose of the present paper is to establish moment estimates of Rosenthal type for a rather general class of random variables satisfying certain bounds on the cumulants. We consider sequences of random variables which satisfy a central…

Let $n$ be a positive integer and $X = [x_{ij}]_{1 \leq i, j \leq n}$ be an $n \times n$\linebreak \noindent sized matrix of independent random variables having joint uniform distribution $$\hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n}…

Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time $S_k$ of a Poisson process with unit rate which is independent of the sequence $\{X_k\}$.…

We solve explicitly the following problem: for a given probability measure mu, we specify a generalised martingale diffusion X which, stopped at an independent exponential time T, is distributed according to mu. The process X is specified…

Let $X_1,X_2,\ldots $ be independent random variables observed sequentially and such that $X_1,\ldots,X_{\theta-1}$ have a common probability density $p_0$, while $X_\theta,X_{\theta+1},\ldots $ are all distributed according to $p_1\neq…

Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for a uniformly random permutation ${\boldsymbol{\pi}}$ of…

Let $\xi_1, \xi_2, \ldots$ be independent copies of a positive random variable $\xi$, $S_0 = 0$, and $S_k = \xi_1+\ldots+\xi_k$, $k \in \mathbb{N}$. Define $N(t) = \inf\{k \in \mathbb{N}: S_k>t\}$ for $t\geq 0$. The process $(N(t))_{t\geq…

The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb {E}[c(X_1,X_2)]$ by varying the joint distribution $(X_1,X_2)$ where the marginal distributions of the random variables $X_1$ and $X_2$ are…

Let $X_1,\ldots,X_N$ be i.i.d.\ random variables distributed like $X$. Suppose that the first $k \geq 3$ moments $\{ \mathbb{E}[X^j] : j = 1,\ldots,k\}$ of $X$ agree with that of the standard Gaussian distribution, that…

Fix some $n \in \mathbb{N}$ and let $X_1, X_2,\dots, X_n$ be independent random variables drawn from the uniform distribution on $[0,1]$. A decision maker is shown the variables sequentially and, after each observation, must decide whether…

There are $n$ independent Bernoulli random variables $I_{k}$ with parameters $p_{k}$ that are observed sequentially. We consider a generalization of the Last-Success-Problem considering $w_{k}$ positive payments if the player successfully…

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of $n$ locations in each time step $t=0,1,2,\dots$. Their goal is to minimize the expected time until they visit the same location and thus…

Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of X_1,...,X_t that contains, as an embedded subsequence, each of the non-order-isomorphic…

Let X be a random variable. We shall call an independent random variable Y to be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is said to be symmetry resistant if the variance of any symmetrizer Y, is never smaller…

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several…

Consider the random graph $G({\mathcal P}_{n},r)$ whose vertex set ${\mathcal P}_{n}$ is a Poisson point process of intensity $n$ on $(- \frac{1}{2}, \frac{1}{2}]^d$, $d \geq 2$. Any two vertices $X_i,X_j \in {\mathcal P}_{n}$ are connected…

We calculate, for a branching random walk $X_n(l)$ to a leaf $l$ at depth $n$ on a binary tree, the positive integer moments of the random variable $\frac{1}{2^{n}}\sum_{l=1}^{2^n}e^{2\beta X_n(l)}$, for $\beta\in\mathbb{R}$. We obtain…