46 research outputs found
Indefinitely Oscillating Martingales
We construct a class of nonnegative martingale processes that oscillate
indefinitely with high probability. For these processes, we state a uniform
rate of the number of oscillations and show that this rate is asymptotically
close to the theoretical upper bound. These bounds on probability and
expectation of the number of upcrossings are compared to classical bounds from
the martingale literature. We discuss two applications. First, our results
imply that the limit of the minimum description length operator may not exist.
Second, we give bounds on how often one can change one's belief in a given
hypothesis when observing a stream of data.Comment: ALT 2014, extended technical repor
Indefinitely oscillating martingales
We construct a class of nonnegative martingale processes that oscillate indefinitely with high probability. For these processes, we state a uniform rate of the number of oscillations for a given magnitude and show that this rate is asymptotically close to the theoretical upper bound. These bounds on probability and expectation of the number of upcrossings are compared to classical bounds from the martingale literature. We discuss two applications. First, our results imply that the limit of the minimum description length operator may not exist. Second, we give bounds on how often one can change oneâs belief in a given hypothesis when observing a stream of data
A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering
Routing mechanisms for stochastic networks are often designed to produce
state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the
limiting process to a lower-dimensional subset of its full state space. In a
fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding
mode control that forces the fluid trajectories to a lower-dimensional sliding
manifold. Within deterministic dynamical systems theory, it is well known that
sliding-mode controls can cause the system to chatter back and forth along the
sliding manifold due to delays in activation of the control. For the prelimit
stochastic system, chattering implies fluid-scaled fluctuations that are larger
than typical stochastic fluctuations. In this paper we show that chattering can
occur in the fluid limit of a controlled stochastic network when inappropriate
control parameters are used. The model has two large service pools operating
under the fixed-queue-ratio with activation and release thresholds (FQR-ART)
overload control which we proposed in a recent paper. We now show that, if the
control parameters are not chosen properly, then delays in activating and
releasing the control can cause chattering with large oscillations in the fluid
limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even
when the arrival rates are smaller than the potential total service rate in the
system, a phenomenon referred to as congestion collapse. We show that the fluid
limit can be a bi-stable switching system possessing a unique nontrivial
periodic equilibrium, in addition to a unique stationary point
Test Martingales, Bayes Factors and -Values
A nonnegative martingale with initial value equal to one measures evidence
against a probabilistic hypothesis. The inverse of its value at some stopping
time can be interpreted as a Bayes factor. If we exaggerate the evidence by
considering the largest value attained so far by such a martingale, the
exaggeration will be limited, and there are systematic ways to eliminate it.
The inverse of the exaggerated value at some stopping time can be interpreted
as a -value. We give a simple characterization of all increasing functions
that eliminate the exaggeration.Comment: Published in at http://dx.doi.org/10.1214/10-STS347 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Simulation of a Brownian particle in an optical trap
Cataloged from PDF version of article.Unlike passive Brownian particles, active Brownian particles, also known as microswimmers, propel themselves with directed motion and thus drive themselves out of equilibrium. Understanding their motion can provide insight into out-of-equilibrium phenomena associated with biological examples such as bacteria, as well as with artificial microswimmers. We discuss how to mathematically model their motion using a set of stochastic differential equations and how to numerically simulate it using the corresponding set of finite difference equations both in homogenous and complex environments. In particular, we show how active Brownian particles do not follow the Maxwell-Boltzmann distribution-a clear signature of their out-of-equilibrium nature- and how, unlike passive Brownian particles, microswimmers can be funneled, trapped, and sorted. (C) 2014 American Association of Physics Teachers
Randomness and early termination: what makes a game exciting?
In this paper we revisit an open problem posed by Aldous on the max-entropy
win-probability martingale: given two players of equal strength, such that the
win-probability is a martingale diffusion, which of these processes has maximum
entropy and hence gives the most excitement for the spectators? We study a
terminal-boundary value problem for the nonlinear parabolic PDE
derived by Aldous and prove its
wellposedness and regularity of its solution by combining PDE analysis and
probabilistic tools, in particular the reformulation as a stochastic control
problem with restricted control set, which allows us to deduce strict
ellipticity. We establish key qualitative properties of the solution including
concavity, monotonicity, convergence to a steady state for long remaining time
and the asymptotic behaviour shortly before the terminal time. Moreover, we
construct convergent numerical approximations. The analytical and numerical
results allow us to highlight the behaviour of the win-probability process in
the present case where the match may end early, in contrast to recent work by
Backhoff-Veraguas and Beiglb\"ock where the match always runs the full length
Offline to Online Conversion
We consider the problem of converting offline estimators into an online
predictor or estimator with small extra regret. Formally this is the problem of
merging a collection of probability measures over strings of length 1,2,3,...
into a single probability measure over infinite sequences. We describe various
approaches and their pros and cons on various examples. As a side-result we
give an elementary non-heuristic purely combinatoric derivation of Turing's
famous estimator. Our main technical contribution is to determine the
computational complexity of online estimators with good guarantees in general.Comment: 20 LaTeX page
Detecting Markov Chain Instability: A Monte Carlo Approach
We devise a Monte Carlo based method for detecting whether a non-negative
Markov chain is stable for a given set of parameter values. More precisely, for
a given subset of the parameter space, we develop an algorithm that is capable
of deciding whether the set has a subset of positive Lebesgue measure for which
the Markov chain is unstable. The approach is based on a variant of simulated
annealing, and consequently only mild assumptions are needed to obtain
performance guarantees.
The theoretical underpinnings of our algorithm are based on a result stating
that the stability of a set of parameters can be phrased in terms of the
stability of a single Markov chain that searches the set for unstable
parameters. Our framework leads to a procedure that is capable of performing
statistically rigorous tests for instability, which has been extensively tested
using several examples of standard and non-standard queueing networks
A flow-based approach to rough differential equations
These are lecture notes for a Master 2 course on rough differential equations
driven by weak geometric Holder p-rough paths, for any p>2. They provide a
short, self-contained and pedagogical account of the theory, with an emphasis
on flows. The theory is illustrated by some now classical applications to
stochastic analysis, such as the basics of Freidlin-Wentzel theory of large
deviations for diffusions, or Stroock and Varadhan support theorem.Comment: 63 page