247 research outputs found
Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable
We give a sufficient condition for the exponential decay of the tail
probability of a non-negative random variable. We consider the
Laplace-Stieltjes transform of the probability distribution function of the
random variable. We present a theorem, according to which if the abscissa of
convergence of the LS transform is negative finite and the real point on the
axis of convergence is a pole of the LS transform, then the tail probability
decays exponentially. For the proof of the theorem, we extend and apply
so-called a finite form of Ikehara's complex Tauberian theorem by
Graham-Vaaler.Comment: 22pages, 1 figure, submitted to IEEE Transactions on Information
Theor
Geometrically stopped Markovian random growth processes and Pareto tails
Many empirical studies document power law behavior in size distributions of
economic interest such as cities, firms, income, and wealth. One mechanism for
generating such behavior combines independent and identically distributed
Gaussian additive shocks to log-size with a geometric age distribution. We
generalize this mechanism by allowing the shocks to be non-Gaussian (but
light-tailed) and dependent upon a Markov state variable. Our main results
provide sharp bounds on tail probabilities, a simple equation determining
Pareto exponents, and comparative statics. We present two applications: we show
that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic
general equilibrium model with idiosyncratic investment risk are Paretian, and
(ii) a random growth model for the population dynamics of Japanese
municipalities is consistent with the observed Pareto exponent but only after
allowing for Markovian dynamics
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On defining partition entropy by inequalities
Partition entropy is the numerical metric of uncertainty within
a partition of a finite set, while conditional entropy measures the degree of
difficulty in predicting a decision partition when a condition partition is
provided. Since two direct methods exist for defining conditional entropy
based on its partition entropy, the inequality postulates of monotonicity,
which conditional entropy satisfies, are actually additional constraints on
its entropy. Thus, in this paper partition entropy is defined as a function
of probability distribution, satisfying all the inequalities of not only partition
entropy itself but also its conditional counterpart. These inequality
postulates formalize the intuitive understandings of uncertainty contained
in partitions of finite sets.We study the relationships between these inequalities,
and reduce the redundancies among them. According to two different
definitions of conditional entropy from its partition entropy, the convenient
and unified checking conditions for any partition entropy are presented, respectively.
These properties generalize and illuminate the common nature
of all partition entropies
Large deviations for the local times of a random walk among random conductances
We derive an annealed large deviation principle for the normalised local
times of a continuous-time random walk among random conductances in a finite
domain in in the spirit of Donsker-Varadhan \cite{DV75}. We work in the
interesting case that the conductances may assume arbitrarily small values.
Thus, the underlying picture of the principle is a joint strategy of small
values of the conductances and large holding times of the walk. The speed and
the rate function of our principle are explicit in terms of the lower tails of
the conductance distribution. As an application, we identify the logarithmic
asymptotics of the lower tails of the principal eigenvalue of the randomly
perturbed negative Laplace operator in the domain.Comment: 12 page
Species lifetime distribution for simple models of ecologies
Interpretation of empirical results based on a taxa's lifetime distribution
shows apparently conflicting results. Species' lifetime is reported to be
exponentially distributed, whereas higher order taxa, such as families or
genera, follow a broader distribution, compatible with power law decay. We show
that both these evidences are consistent with a simple evolutionary model that
does not require specific assumptions on species interaction. The model
provides a zero-order description of the dynamics of ecological communities and
its species lifetime distribution can be computed exactly. Different behaviors
are found: an initial power law, emerging from a random walk type of
dynamics, which crosses over to a steeper branching process-like
regime and finally is cutoff by an exponential decay which becomes weaker and
weaker as the total population increases. Sampling effects can also be taken
into account and shown to be relevant: if species in the fossil record were
sampled according to the Fisher log-series distribution, lifetime should be
distributed according to a power law. Such variability of behaviors in
a simple model, combined with the scarcity of data available, cast serious
doubts on the possibility to validate theories of evolution on the basis of
species lifetime data.Comment: 19 pages, 2 figure
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