1,645 research outputs found
Chaoticity for multi-class systems and exchangeability within classes
Classical results for exchangeable systems of random variables are extended
to multi-class systems satisfying a natural partial exchangeability assumption.
It is proved that the conditional law of a finite multi-class system, given the
value of the vector of the empirical measures of its classes, corresponds to
independent uniform orderings of the samples within each class, and that a
family of such systems converges in law if and only if the corresponding
empirical measure vectors converge in law. As a corollary, convergence within
each class to an infinite i.i.d. system implies asymptotic independence between
different classes. A result implying the Hewitt-Savage 0-1 Law is also
extended.Comment: Third revision, v4. The paper is similar to the second revision v3,
with several improvement
Regenerative properties of the linear hawkes process with unbounded memory
We prove regenerative properties for the linear Hawkes process under minimal
assumptions on the transfer function, which may have unbounded support. These
results are applicable to sliding window statistical estimators. We exploit
independence in the Poisson cluster point process decomposition, and the
regeneration times are not stopping times for the Hawkes process. The
regeneration time is interpreted as the renewal time at zero of a M/G/infinity
queue, which yields a formula for its Laplace transform. When the transfer
function admits some exponential moments, we stochastically dominate the
cluster length by exponential random variables with parameters expressed in
terms of these moments. This yields explicit bounds on the Laplace transform of
the regeneration time in terms of simple integrals or special functions
yielding an explicit negative upper-bound on its abscissa of convergence. These
regenerative results allow, e.g., to systematically derive long-time asymptotic
results in view of statistical applications. This is illustrated on a
concentration inequality previously obtained with coauthors
Out of equilibrium functional central limit theorems for a large network where customers join the shortest of several queues
Customers arrive at rate N times alpha on a network of N single server
infinite buffer queues, choose L queues uniformly, join the shortest one, and
are served there in turn at rate beta. We let N go to infinity.We prove a
functional central limit theorem (CLT) for the tails of the empirical measures
of the queue occupations,in a Hilbert space with the weak topology, with limit
given by an Ornstein-Uhlenbeck process. The a priori assumption is that the
initial data converge.This completes a recent functional CLT in equilibrium
result for which convergence for the initial data was not known in advance, but
was deduced a posteriori from the functional CLT.Comment: A new preprint math.PR/0403538, has been written as a combined
version of the present preprint and the preprint math.PR/0312334. It is
recommended to read the new combined version instead of the two other
Interacting multi-class transmissions in large stochastic networks
The mean-field limit of a Markovian model describing the interaction of
several classes of permanent connections in a network is analyzed. Each of the
connections has a self-adaptive behavior in that its transmission rate along
its route depends on the level of congestion of the nodes of the route. Since
several classes of connections going through the nodes of the network are
considered, an original mean-field result in a multi-class context is
established. It is shown that, as the number of connections goes to infinity,
the behavior of the different classes of connections can be represented by the
solution of an unusual nonlinear stochastic differential equation depending not
only on the sample paths of the process, but also on its distribution.
Existence and uniqueness results for the solutions of these equations are
derived. Properties of their invariant distributions are investigated and it is
shown that, under some natural assumptions, they are determined by the
solutions of a fixed-point equation in a finite-dimensional space.Comment: Published in at http://dx.doi.org/10.1214/09-AAP614 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Self-adaptive congestion control for multi-class intermittent connections in a communication network
A Markovian model of the evolution of intermittent connections of various
classes in a communication network is established and investigated. Any
connection evolves in a way which depends only on its class and the state of
the network, in particular as to the route it uses among a subset of the
network nodes. It can be either active (ON) when it is transmitting data along
its route, or idle (OFF). The congestion of a given node is defined as a
functional of the transmission rates of all ON connections going through it,
and causes losses and delays to these connections. In order to control this,
the ON connections self-adaptively vary their transmission rate in TCP-like
fashion. The connections interact through this feedback loop. A Markovian model
is provided by the states (OFF, or ON with some transmission rate) of the
connections. The number of connections in each class being potentially huge, a
mean-field limit result is proved with an appropriate scaling so as to reduce
the dimensionality. In the limit, the evolution of the states of the
connections can be represented by a non-linear system of stochastic
differential equations, of dimension the number of classes. Additionally, it is
shown that the corresponding stationary distribution can be expressed by the
solution of a fixed-point equation of finite dimension
Cache Miss Estimation for Non-Stationary Request Processes
The aim of the paper is to evaluate the miss probability of a Least Recently
Used (LRU) cache, when it is offered a non-stationary request process given by
a Poisson cluster point process. First, we construct a probability space using
Palm theory, describing how to consider a tagged document with respect to the
rest of the request process. This framework allows us to derive a general
integral formula for the expected number of misses of the tagged document.
Then, we consider the limit when the cache size and the arrival rate go to
infinity proportionally, and use the integral formula to derive an asymptotic
expansion of the miss probability in powers of the inverse of the cache size.
This enables us to quantify and improve the accuracy of the so-called Che
approximation
The Bounded Confidence Model Of Opinion Dynamics
The bounded confidence model of opinion dynamics, introduced by Deffuant et
al, is a stochastic model for the evolution of continuous-valued opinions
within a finite group of peers. We prove that, as time goes to infinity, the
opinions evolve globally into a random set of clusters too far apart to
interact, and thereafter all opinions in every cluster converge to their
barycenter. We then prove a mean-field limit result, propagation of chaos: as
the number of peers goes to infinity in adequately started systems and time is
rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov
(or McKean-Vlasov) processes; the limit opinion processes evolves as if under
the influence of opinions drawn from its own instantaneous law, which are the
unique solution of a nonlinear integro-differential equation of Kac type. This
implies that the (random) empirical distribution processes converges to this
(deterministic) solution. We then prove that, as time goes to infinity, this
solution converges to a law concentrated on isolated opinions too far apart to
interact, and identify sufficient conditions for the limit not to depend on the
initial condition, and to be concentrated at a single opinion. Finally, we
prove that if the equation has an initial condition with a density, then its
solution has a density at all times, develop a numerical scheme for the
corresponding functional equation, and show numerically that bifurcations may
occur.Comment: 43 pages, 7 figure
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