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| Speaker |
Mark Brown, The City College, CUNY |
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| Title |
The Waiting Time for Patterns |
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| Abstract | The following are examples of pattern problems..
1) How long will it take for an 80% free-throw shooter to make 20
consecutive foul shots?. This is an example of the problem of
"success runs."
2) In rolls of a fair die how long will it take to get m consecutive
identical outcomes, of any kind?. This is the minimum of 6 dependent
success run times. We refer to it as multi-type success runs.
3) How long will it take for a random number generator to produce a
consecutive string of 5 digits where the digits 1,2,3,4,5 appear once
each in any order?. We refer to that as the waiting time for
permutations. We can also consider cyclic permutations.
4) How long will it take for the oft-mentioned monkey randomly banging
on a typewriter to type out Hamlet?
The waiting time for patterns has interested many authors, going back
at least to Von-Mises in 1912. The mean waiting time is a favorite of
probability texts. For example many people are surprised to learn that
in flips of a fair coin the expected waiting time for HHH equals 14,
for HTH equals 10, and for HHT equals 8.
We derive approximations for the waiting time distribution, along with
bounds on the total variation distance between the approximate and
true distribution. Our approach is based on generating functions and
hazard rate inequalities. The results improve upon those obtained by
Chen-Stein methodology, in several respects.
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| Speaker |
Michael Marcus, CUNY/City College |
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| Title |
Strong Laws for Gaussian Differences |
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| Abstract |
Using the Borell, Sudakov-Tsirelson Theorem we obtain a
surprising, robust, strong law of large numbers for the LP norm of
integrated Gaussian differences. This is a preliminary report on joint
work with Professor Jay Rosen |
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| Speaker |
Scott Sheffield, Courant |
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| Title |
Tug of war and the infinity Laplacian |
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| Abstract |
The infinity Laplacian (informally, the ßecond derivative in the
gradient direction") is a simple yet mysterious operator with many
applications.
"Tug of war" is a two player random turn game played as follows:
SETUP: Assign each player one of two disjoint target sets T1 and T2
in the plane, and fix a starting position x and a constant e.
Place the game token at x.
GAME PLAY: Toss a fair coin and allow the player who wins the coin
toss to move the game token up to epsilon units in the direction of
his or her choice. Repeat the above until the token reaches a target
set Ti. The ith player is then declared the winner.
Given parameters e and x, write ue(x) for the
probability that player one wins when both players play optimally. We
show that as epsilon tends to zero, the functions ue(x)
converge to the infinity harmonic function with boundary conditions 1
on T1 and 0 on T2.
Our strategic analysis of tug of war enables us to solve the öptimal
Lipschitz extension problem" and leads to new formulations and
significant generalizations of several classical results about
infinity laplacians. The game theoretic arguments are simpler and
more elementary than the original proofs.
This talk is based on joint work with Yuval Peres, Oded Schramm, and
David Wilson. |
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| Speaker |
Pavel Hitzcenko, Drexel University |
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| Title |
Probabilistic analysis of a class of WHT algorithms. |
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| Abstract |
The Walsh-Hadamard transform (WHT) is one of the frequently
used transforms in signal processing. Since its computation typically
involves large data, a significant effort was put into developing
efficient algorithms for its computation and into construction of
theoretical models that would explain empirical observations. In this
talk, after briefly describing a family of algorithms based on a
factorization of a WHT matrix, I will discuss probabilistic aspects of
one of the important measures of their complexity, namely the
instruction count.
The talk is based on a joint work with J. Johnson(CS, Drexel) and his
(former) student H-J. Huang.
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| Speaker |
Makram Talih, Hunter College of the City University of New York |
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| Title |
Strategies for Online Inference in Dynamic Graphical Models |
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| Abstract | We present different strategies for online learning of
Dynamic Graphs (DG's) via Sequential Monte Carlo (SMC). SMC
algorithms are based on maintaining in parameter space an ensemble of
particles, each of which tracks a particular realization of the
process under study. In our framework, the process is governed by the
posterior distribution of the parameters (eg. the precision matrix)
and hidden variables (eg. the underlying undirected graph) given the
data sequence. |
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| Speaker |
Soumik Pal, Columbia University |
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| Title |
Capital Requirement to Achieve Acceptability |
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| Abstract |
Consider an agent who wishes to trade in a financial market
between day t=0 and day t=T. The agent starts with a specific
objective in mind, which defines a class of acceptable or riskless
(random) financial positions at the end of day T. It has been shown
in the theory of convex measures of risk that any "good" definition
of acceptability will stipulate a random variable to be acceptable if
its expectation under a variety of probability measures (called
scenarios) dominate a corresponding real number (called floors). By
suitable choices of scenarios and floors, this simple structure
becomes flexible to encompass several major problems in mathematical
finance, including superhedging, efficient hedging, and robust
trading.
In this talk we find, under general modeling assumptions, the minimum
initial capital, the agent would require to drive his portfolio
towards acceptability. The solution leads to a beautiful mathematical
picture, which ties its analytical side (Frechet derivatives in
Hilbert spaces), geometric side (nearest point projections in
uniformly convex Banach spaces), and how such analytic and geometric
properties become amenable under probabilistic structures of the
Brownian filtration.
In specific cases, such minimum initial capital becomes sellers'
price for efficient hedging on the one hand, and no-good-deals
valuation on the other hand. And we shall see examples in this
direction. |
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