27 research outputs found
On an Asymptotic Series of Ramanujan
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is
concerned with the behavior of the expected value of for large
where is a Poisson random variable with mean and
is a function satisfying certain growth conditions. We generalize this by
studying the asymptotics of the expected value of when the
distribution of belongs to a suitable family indexed by a convolution
parameter. Examples include the problem of inverse moments for distribution
families such as the binomial or the negative binomial.Comment: To appear, Ramanujan
Partial fillup and search time in LC tries
Andersson and Nilsson introduced in 1993 a level-compressed trie (in short:
LC trie) in which a full subtree of a node is compressed to a single node of
degree being the size of the subtree. Recent experimental results indicated a
'dramatic improvement' when full subtrees are replaced by partially filled
subtrees. In this paper, we provide a theoretical justification of these
experimental results showing, among others, a rather moderate improvement of
the search time over the original LC tries. For such an analysis, we assume
that n strings are generated independently by a binary memoryless source with p
denoting the probability of emitting a 1. We first prove that the so called
alpha-fillup level (i.e., the largest level in a trie with alpha fraction of
nodes present at this level) is concentrated on two values with high
probability. We give these values explicitly up to O(1), and observe that the
value of alpha (strictly between 0 and 1) does not affect the leading term.
This result directly yields the typical depth (search time) in the alpha-LC
tries with p not equal to 1/2, which turns out to be C loglog n for an
explicitly given constant C (depending on p but not on alpha). This should be
compared with recently found typical depth in the original LC tries which is C'
loglog n for a larger constant C'. The search time in alpha-LC tries is thus
smaller but of the same order as in the original LC tries.Comment: 13 page
On the asymptotic behavior of some Algorithms
A simple approach is presented to study the asymptotic behavior of some
algorithms with an underlying tree structure. It is shown that some asymptotic
oscillating behaviors can be precisely analyzed without resorting to complex
analysis techniques as it is usually done in this context. A new explicit
representation of periodic functions involved is obtained at the same time.Comment: November 200
Sharp Bounds on the Entropy of the Poisson Law and Related Quantities
One of the difficulties in calculating the capacity of certain Poisson
channels is that H(lambda), the entropy of the Poisson distribution with mean
lambda, is not available in a simple form. In this work we derive upper and
lower bounds for H(lambda) that are asymptotically tight and easy to compute.
The derivation of such bounds involves only simple probabilistic and analytic
tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and
Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on
the relative entropy D(n, p) between a binomial and a Poisson, thus refining
the work of Harremoes and Ruzankin (2004). Bounds on the entropy of the
binomial also follow easily.Comment: To appear, IEEE Trans. Inform. Theor
Direct Estimation of Information Divergence Using Nearest Neighbor Ratios
We propose a direct estimation method for R\'{e}nyi and f-divergence measures
based on a new graph theoretical interpretation. Suppose that we are given two
sample sets and , respectively with and samples, where
is a constant value. Considering the -nearest neighbor (-NN)
graph of in the joint data set , we show that the average powered
ratio of the number of points to the number of points among all -NN
points is proportional to R\'{e}nyi divergence of and densities. A
similar method can also be used to estimate f-divergence measures. We derive
bias and variance rates, and show that for the class of -H\"{o}lder
smooth functions, the estimator achieves the MSE rate of
. Furthermore, by using a weighted ensemble
estimation technique, for density functions with continuous and bounded
derivatives of up to the order , and some extra conditions at the support
set boundary, we derive an ensemble estimator that achieves the parametric MSE
rate of . Our estimators are more computationally tractable than other
competing estimators, which makes them appealing in many practical
applications.Comment: 2017 IEEE International Symposium on Information Theory (ISIT
On Space-Time Capacity Limits in Mobile and Delay Tolerant Networks
We investigate the fundamental capacity limits of space-time journeys of
information in mobile and Delay Tolerant Networks (DTNs), where information is
either transmitted or carried by mobile nodes, using store-carry-forward
routing. We define the capacity of a journey (i.e., a path in space and time,
from a source to a destination) as the maximum amount of data that can be
transferred from the source to the destination in the given journey. Combining
a stochastic model (conveying all possible journeys) and an analysis of the
durations of the nodes' encounters, we study the properties of journeys that
maximize the space-time information propagation capacity, in bit-meters per
second. More specifically, we provide theoretical lower and upper bounds on the
information propagation speed, as a function of the journey capacity. In the
particular case of random way-point-like models (i.e., when nodes move for a
distance of the order of the network domain size before changing direction), we
show that, for relatively large journey capacities, the information propagation
speed is of the same order as the mobile node speed. This implies that,
surprisingly, in sparse but large-scale mobile DTNs, the space-time information
propagation capacity in bit-meters per second remains proportional to the
mobile node speed and to the size of the transported data bundles, when the
bundles are relatively large. We also verify that all our analytical bounds are
accurate in several simulation scenarios.Comment: Part of this work will be presented in "On Space-Time Capacity Limits
in Mobile and Delay Tolerant Networks", P. Jacquet, B. Mans and G. Rodolakis,
IEEE Infocom, 201
On sequential selection and a first passage problem for the Poisson process
This note is motivated by connections between the online and offline problems
of selecting a possibly long subsequence from a Poisson-paced sequence of
uniform marks under either a monotonicity or a sum constraint. The offline
problem with the sum constraint amounts to counting the Poisson arrivals before
their total exceeds a certain level. A precise asymptotics for the mean count
is obtained by coupling with a nonlinear pure birth process