1,604 research outputs found
Information Geometry Approach to Parameter Estimation in Markov Chains
We consider the parameter estimation of Markov chain when the unknown
transition matrix belongs to an exponential family of transition matrices.
Then, we show that the sample mean of the generator of the exponential family
is an asymptotically efficient estimator. Further, we also define a curved
exponential family of transition matrices. Using a transition matrix version of
the Pythagorean theorem, we give an asymptotically efficient estimator for a
curved exponential family.Comment: Appendix D is adde
Joint Beamforming and Power Control in Coordinated Multicell: Max-Min Duality, Effective Network and Large System Transition
This paper studies joint beamforming and power control in a coordinated
multicell downlink system that serves multiple users per cell to maximize the
minimum weighted signal-to-interference-plus-noise ratio. The optimal solution
and distributed algorithm with geometrically fast convergence rate are derived
by employing the nonlinear Perron-Frobenius theory and the multicell network
duality. The iterative algorithm, though operating in a distributed manner,
still requires instantaneous power update within the coordinated cluster
through the backhaul. The backhaul information exchange and message passing may
become prohibitive with increasing number of transmit antennas and increasing
number of users. In order to derive asymptotically optimal solution, random
matrix theory is leveraged to design a distributed algorithm that only requires
statistical information. The advantage of our approach is that there is no
instantaneous power update through backhaul. Moreover, by using nonlinear
Perron-Frobenius theory and random matrix theory, an effective primal network
and an effective dual network are proposed to characterize and interpret the
asymptotic solution.Comment: Some typos in the version publised in the IEEE Transactions on
Wireless Communications are correcte
Renewal theorems for a class of processes with dependent interarrival times and applications in geometry
Renewal theorems are developed for point processes with interarrival times
, where is a stochastic
process with finite state space and is
a H\"older continuous function on a subset .
The theorems developed here unify and generalise the key renewal theorem for
discrete measures and Lalley's renewal theorem for counting measures in
symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The
new renewal theorems allow for direct applications to problems in fractal and
hyperbolic geometry; for instance, results on the Minkowski measurability of
self-conformal sets are deduced. Indeed, these geometric problems motivated the
development of the renewal theorems.Comment: 2 figure
Vibrational modes and spectrum of oscillators on a scale-free network
We study vibrational modes and spectrum of a model system of atoms and
springs on a scale-free network in order to understand the nature of
excitations with many degrees of freedom on the scale-free network. We assume
that the atoms and springs are distributed as nodes and links of a scale-free
network, assigning the mass and the specific oscillation frequency
of the -th atom and the spring constant between the
-th and -th atoms.Comment: 8pages, 2 figure
Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets
We show that the fractal curvature measures of invariant sets of
one-dimensional conformal iterated function systems satisfying the open set
condition exist, if and only if the associated geometric potential function is
nonlattice. Moreover, in the nonlattice situation we obtain that the Minkowski
content exists and prove that the fractal curvature measures are constant
multiples of the -conformal measure, where denotes the
Minkowski dimension of the invariant set. For the first fractal curvature
measure, this constant factor coincides with the Minkowski content of the
invariant set. In the lattice situation we give sufficient conditions for the
Minkowski content of the invariant set to exist, contrasting the fact that the
Minkowski content of a self-similar lattice fractal never exists. However,
every self-similar set satisfying the open set condition exhibits a Minkowski
measurable diffeomorphic image. Both in the lattice
and nonlattice situation average versions of the fractal curvature measures are
shown to always exist.Comment: 36 page
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