306 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
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
Tight Markov chains and random compositions
For an ergodic Markov chain on , with a stationary
distribution , let denote a hitting time for , and let
. Around 2005 Guy Louchard popularized a conjecture that, for , is almost Geometric(), , is almost
stationarily distributed on , and that and are almost
independent, if exponentially fast. For the
chains with however slowly, and with
, we show that Louchard's
conjecture is indeed true even for the hits of an arbitrary
with . More precisely, a sequence of consecutive hit
locations paired with the time elapsed since a previous hit (for the first hit,
since the starting moment) is approximated, within a total variation distance
of order , by a -long sequence of independent copies of
, where , is
distributed stationarily on , and is independent of . The
two conditions are easily met by the Markov chains that arose in Louchard's
studies as likely sharp approximations of two random compositions of a large
integer , a column-convex animal (cca) composition and a Carlitz (C)
composition. We show that this approximation is indeed very sharp for most of
the parts of the random compositions. Combining the two approximations in a
tandem, we are able to determine the limiting distributions of
and largest parts of the random cca composition and the
random C-composition, respectively. (Submitted to Annals of Probability in
August, 2009.
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
Cooperative learning in multi-agent systems from intermittent measurements
Motivated by the problem of tracking a direction in a decentralized way, we
consider the general problem of cooperative learning in multi-agent systems
with time-varying connectivity and intermittent measurements. We propose a
distributed learning protocol capable of learning an unknown vector from
noisy measurements made independently by autonomous nodes. Our protocol is
completely distributed and able to cope with the time-varying, unpredictable,
and noisy nature of inter-agent communication, and intermittent noisy
measurements of . Our main result bounds the learning speed of our
protocol in terms of the size and combinatorial features of the (time-varying)
networks connecting the nodes
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