183 research outputs found
New results on a generalized coupon collector problem using Markov chains
We study in this paper a generalized coupon collector problem, which consists
in determining the distribution and the moments of the time needed to collect a
given number of distinct coupons that are drawn from a set of coupons with an
arbitrary probability distribution. We suppose that a special coupon called the
null coupon can be drawn but never belongs to any collection. In this context,
we obtain expressions of the distribution and the moments of this time. We also
prove that the almost-uniform distribution, for which all the non-null coupons
have the same drawing probability, is the distribution which minimizes the
expected time to get a fixed subset of distinct coupons. This optimization
result is extended to the complementary distribution of that time when the full
collection is considered, proving by the way this well-known conjecture.
Finally, we propose a new conjecture which expresses the fact that the
almost-uniform distribution should minimize the complementary distribution of
the time needed to get any fixed number of distinct coupons.Comment: 14 page
Optimization results for a generalized coupon collector problem
We study in this paper a generalized coupon collector problem, which consists
in analyzing the time needed to collect a given number of distinct coupons that
are drawn from a set of coupons with an arbitrary probability distribution. We
suppose that a special coupon called the null coupon can be drawn but never
belongs to any collection. In this context, we prove that the almost uniform
distribution, for which all the non-null coupons have the same drawing
probability, is the distribution which stochastically minimizes the time needed
to collect a fixed number of distinct coupons. Moreover, we show that in a
given closed subset of probability distributions, the distribution with all its
entries, but one, equal to the smallest possible value is the one, which
stochastically maximizes the time needed to collect a fixed number of distinct
coupons. An computer science application shows the utility of these results.Comment: arXiv admin note: text overlap with arXiv:1402.524
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement
This paper studies Markov chains on the chambers of real hyperplane
arrangements, a model that generalizes famous examples, such as the Tsetlin
library and riffle shuffles. We discuss cutoff for the Tsetlin library for
general weights, and we give an exact formula for the separation distance for
the hyperplane arrangement walk. We introduce lower bounds, which allow for the
first time to study cutoff for hyperplane arrangement walks under certain
conditions. Using similar techniques, we also prove a uniform lower bound for
the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833
Markov chains, -trivial monoids and representation theory
We develop a general theory of Markov chains realizable as random walks on
-trivial monoids. It provides explicit and simple formulas for the
eigenvalues of the transition matrix, for multiplicities of the eigenvalues via
M\"obius inversion along a lattice, a condition for diagonalizability of the
transition matrix and some techniques for bounding the mixing time. In
addition, we discuss several examples, such as Toom-Tsetlin models, an exchange
walk for finite Coxeter groups, as well as examples previously studied by the
authors, such as nonabelian sandpile models and the promotion Markov chain on
posets. Many of these examples can be viewed as random walks on quotients of
free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth
birthday; 71 pages; final version to appear in IJA
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