412 research outputs found
The perimeter of uniform and geometric words: a probabilistic analysis
Let a word be a sequence of i.i.d. integer random variables. The
perimeter of the word is the number of edges of the word, seen as a
polyomino. In this paper, we present a probabilistic approach to the
computation of the moments of . This is applied to uniform and geometric
random variables. We also show that, asymptotically, the distribution of is
Gaussian and, seen as a stochastic process, the perimeter converges in
distribution to a Brownian motionComment: 13 pages, 7 figure
A refined and asymptotic analysis of optimal stopping problems of Bruss and Weber
The classical secretary problem has been generalized over the years into
several directions. In this paper we confine our interest to those
generalizations which have to do with the more general problem of stopping on a
last observation of a specific kind. We follow Dendievel, (where a bibliography
can be found) who studies several types of such problems, mainly initiated by
Bruss and Weber. Whether in discrete time or continuous time, whether all
parameters are known or must be sequentially estimated, we shall call such
problems simply "Bruss-Weber problems". Our contribution in the present paper
is a refined analysis of several problems in this class and a study of the
asymptotic behaviour of solutions.
The problems we consider center around the following model. Let
be a sequence of independent random variables which can
take three values: Let p:=\P(X_i=1), p':=\P(X_i=-1),
\qt:=\P(X_i=0), p\geq p', where p+p'+\qt=1. The goal is to maximize the
probability of stopping on a value or appearing for the last time in
the sequence. Following a suggestion by Bruss, we have also analyzed an
x-strategy with incomplete information: the cases known, unknown, then
known, unknown and finally unknown are considered. We also
present simulations of the corresponding complete selection algorithm.Comment: 22 pages, 19 figure
Tail estimates for the Brownian excursion area and other Brownian areas
Several Brownian areas are considered in this paper: the Brownian excursion
area, the Brownian bridge area, the Brownian motion area, the Brownian meander
area, the Brownian double meander area, the positive part of Brownian bridge
area, the positive part of Brownian motion area. We are interested in the
asymptotics of the right tail of their density function. Inverting a double
Laplace transform, we can derive, in a mechanical way, all terms of an
asymptotic expansion. We illustrate our technique with the computation of the
first four terms. We also obtain asymptotics for the right tail of the
distribution function and for the moments. Our main tool is the two-dimensional
saddle point method.Comment: 34 page
The Adaptive Sampling Revisited
The problem of estimating the number of distinct keys of a large
collection of data is well known in computer science. A classical algorithm
is the adaptive sampling (AS). can be estimated by , where is
the final bucket (cache) size and is the final depth at the end of the
process. Several new interesting questions can be asked about AS (some of them
were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution
of is known, we rederive this distribution in a simpler way.
We provide new results on the moments of and . We also analyze the final
cache size distribution. We consider colored keys: assume that among the
distinct keys, do have color . We show how to estimate
. We also study colored keys with some multiplicity given by
some distribution function. We want to estimate mean an variance of this
distribution. Finally, we consider the case where neither colors nor
multiplicities are known. There we want to estimate the related parameters. An
appendix is devoted to the case where the hashing function provides bits with
probability different from
The maximum of Brownian motion with parabolic drift
We study the maximum of a Brownian motion with a parabolic drift; this is a
random variable that often occurs as a limit of the maximum of discrete
processes whose expectations have a maximum at an interior point. We give
series expansions and integral formulas for the distribution and the first two
moments, together with numerical values to high precision.Comment: 37 page
Monotone runs of uniformly distributed integer random variables: A probabilistic analysis
AbstractUsing a Markov chain approach and a polyomino-like description, we study some asymptotic properties of monotone increasing runs of uniformly distributed integer random variables. We analyze the limiting trajectories, which after suitable normalization, lead to a Brownian motion, the number of runs, which is asymptotically Gaussian, the run length distribution, the hitting time to a large length k run, which is asymptotically exponential, and the maximum run length which is related to the Gumbel extreme-value distribution function. A preliminary application to DNA analysis is also given
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