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For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M blocks such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X_n converges weakly to a distribution that we can describe precisely. We then prove that sup Pr(X_n <= Y_n) converges to a constant, where the supremum is taken over all possible couplings of X_n and Y_n. This constant is expressed explicitly in terms of the parameters of the system.
Place an obstacle with probability $1-p$ independently at each vertex of $mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d geq 2$ and $p$ strictly above the critical threshold for site p
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $mathbb{Z}^d$, $dgeq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asympt
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation,
A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schlafli random cone of a random conical tessellation generated by $n$ independent an
A definition of $d$--dimensional $n$--Meixner random vectors is given first. This definition involves the commutators of their semi--quantum operators. After that we will focus on the $1$-Meixner random vectors, and derive a system of $d$ partial dif