Supremum of a Process in Terms of Trees (original) (raw)

Electronic Journal of Probability, 2011

Let T be a tree with induced partial order . We investigate centered Gaussian processes X = (X t ) t∈T represented as

A Daniell-Kolmogorov theorem for supremum preserving upper probabilities

Fuzzy Sets and Systems, 1999

Possibility measures are interpreted as upper probabilities that are in particular supremum preserving. We define a possibilistic process as a special family of possibilistic variables, and show how its possibility distribution functions can be constructed. We introduce and study the notions of inner and outer regularity for possibility measures. Using these notions, we prove an analogon for possibilistic processes (and possibility measures) of the well-known probabilistic Daniell-Kolmogorov theorem, in the important special case that the variables assume values in compact spaces, and that the possibility measures involved are regular.

A note on the hitting probabilities of random covering sets

Annales Academiae Scientiarum Fennicae Mathematica, 2014

Let E = lim sup n→∞ (g n + ξ n) be the random covering set on the torus T d , where {g n } is a sequence of ball-like sets and ξ n is a sequence of independent random variables uniformly distributed on T d. We prove that E ∩ F = ∅ almost surely whenever F ⊂ T d is an analytic set with Hausdorff dimension, dim H (F) > d − α, where α is the almost sure Hausdorff dimension of E. Moreover, examples are given to show that the condition on dim H (F) cannot be replaced by the packing dimension of F .

The positive entropy kernel for some families of trees

Nonlinearity, 2007

The fact that a continuous self-map of a tree has positive topological entropy is related to the amount of different gods (greatest odd divisors) exhibited by its set of periods. Llibre & Misiurewicz [11] and Blokh [9] give generic upper bounds for the maximum number of gods that a zero entropy tree map f : T −→ T can exhibit, in terms of the number of endpoints and edges of T. In this paper we compute exactly the minimum of the positive integers n such that the entropy of each tree map f : T −→ T exhibiting more than n gods is necessarily positive, for the family of trees which have a subinterval containing all the branching points (this family includes the interval and the stars). We also compute which gods are admissible for such maps.

3 a Note on the Hitting Probabilities of Random Covering Sets

2016

A local maximal inequality under uniform entropy

2011

Abstract We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.

Hausdorff dimensions and Hitting probabilities for some general Gaussian processes

2021

Let B be a d-dimensional Gaussian process on R, where the component are independents copies of a scalar Gaussian process B0 on R+ with a given general variance function γ (r) = Var (B0(r)) and a canonical metric δ(t, s) := (E (B0(t)−B0(s))2) which is commensurate with γ(t − s). We provide some general condition on γ so that for any Borel set E ⊂ [0, 1], the Hausdorff dimension of the image B(E) is constant a.s., and we explicit this constant. Also, we derive under some mild assumptions on γ an upper and lower bounds of P {B(E) ∩ F 6= ∅} in terms of the corresponding Hausdorff measure and capacity of E × F . Some upper and lower bounds for the essential supremum norm of the Hausdorff dimension of B(E) ∩ F and E ∩B−1(F ) are also given in terms of d and the corresponding Hausdorff dimensions of E × F , E, and F .

Threshold Progressions in a Variety of Covering and Packing Contexts

arXiv: Combinatorics, 2018

Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects calA{\cal A}calA being "covered" by at most one object in a random collection calC{\cal C}calC, to being covered at most lambda\lambdalambda times, to being covered at least once, to being covered at least lambda\lambdalambda times, a hierarchy of thresholds emerge. We will then see how such results vary according to the context, and level of dependence introduced. Examples will be from extremal set theory, combinatorics, and additive number theory.

On the profile of random trees

Random Structures and Algorithms, 1997

An approach via generating functions is used to derive multivariate asymptotic distributions for the number of nodes in strata of random forests. For a certain range for the strata numbers we obtain a weak limit theorem to Brownian motion as well. Moreover, a moment convergence theorem for the width of random forests is derived.

Supremum of a Process in Terms of Trees (original) (raw)

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