189 research outputs found
Continued fractions built from convex sets and convex functions
In a partially ordered semigroup with the duality (or polarity) transform, it
is possible to define a generalisation of continued fractions. General
sufficient conditions for convergence of continued fractions with deterministic
terms are provided. Two particular applications concern the cases of convex
sets with the Minkowski addition and the polarity transform (where also
necessary and sufficient conditions of convergence for continued fractions with
constant terms are obtained) and the family of non-negative convex functions
with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.Comment: 18 pages. This version deals with the deterministic case only and is
due to appear in Communications in Contemporary Mathematics. The random case
will be posted separatel
Band depths based on multiple time instances
Bands of vector-valued functions are defined by
considering convex hulls generated by their values concatenated at
different values of the argument. The obtained -bands are families of
functions, ranging from the conventional band in case the time points are
individually considered (for ) to the convex hull in the functional space
if the number of simultaneously considered time points becomes large enough
to fill the whole time domain. These bands give rise to a depth concept that is
new both for real-valued and vector-valued functions.Comment: 12 page
Limit theorems for multidimensional renewal sets
Consider multiple sums on the -dimensional integer grid,which are
generated by i.i.d.\ random variables with a positive expectation. We prove the
strong law of large numbers, the law of the iterated logarithm and the
distributional limit theorem for random sets that appear as
inversion of the multiple sums, that is, as the set of all arguments
such that the interpolated multiple sum exceeds
. The moment conditions are identical to those imposed in the almost sure
limit theorems for multiple sums. The results are expressed in terms of set
inclusions and using distances between sets.Comment: 24 pages. The results are extended to the lower limit in the law of
the iterated logarith
Convex Hulls of L\'evy Processes
Let , , be a L\'evy process in starting at the
origin. We study the closed convex hull of . In
particular, we provide conditions for the integrability of the intrinsic
volumes of the random set and find explicit expressions for their means
in the case of symmetric -stable L\'evy processes. If the process is
symmetric and each its one-dimensional projection is non-atomic, we establish
that the origin a.s. belongs to the interior of for all . Limit
theorems for the convex hull of L\'evy processes with normal and stable limits
are also obtained.Comment: 11 page
A generalisation of the fractional Brownian field based on non-Euclidean norms
We explore a generalisation of the L\'evy fractional Brownian field on the
Euclidean space based on replacing the Euclidean norm with another norm. A
characterisation result for admissible norms yields a complete description of
all self-similar Gaussian random fields with stationary increments. Several
integral representations of the introduced random fields are derived. In a
similar vein, several non-Euclidean variants of the fractional Poisson field
are introduced and it is shown that they share the covariance structure with
the fractional Brownian field and converge to it. The shape parameters of the
Poisson and Brownian variants are related by convex geometry transforms, namely
the radial th mean body and the polar projection transforms.Comment: 28 pages, To appear in J. Math. Anal. App
Sieving random iterative function systems
It is known that backward iterations of independent copies of a contractive
random Lipschitz function converge almost surely under mild assumptions. By a
sieving (or thinning) procedure based on adding to the functions time and space
components, it is possible to construct a scale invariant stochastic process.
We study its distribution and paths properties. In particular, we show that it
is c\`adl\`ag and has finite total variation. We also provide examples and
analyse various properties of particular sieved iterative function systems
including perpetuities and infinite Bernoulli convolutions, iterations of
maximum, and random continued fractions.Comment: 36 pages, 2 figures; accepted for publication in Bernoull
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