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Simple Monte Carlo integration with respect to Bernoulli convolutions

David M. Gómez, Pablo Dartnell (2012)

Applications of Mathematics

We apply a Markov chain Monte Carlo method to approximate the integral of a continuous function with respect to the asymmetric Bernoulli convolution and, in particular, with respect to a binomial measure. This method---inspired by a cognitive model of memory decay---is extremely easy to implement, because it samples only Bernoulli random variables and combines them in a simple way so as to obtain a sequence of empirical measures converging almost surely to the Bernoulli convolution. We give explicit...

Singular measures and the key of G.

Stephen M. Buckley, Paul MacManus (2000)

Publicacions Matemàtiques

We construct a sequence of doubling measures, whose doubling constants tend to 1, all for which kill a Gδ set of full Lebesgue measure.

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition ( T ' ) . We show that for every ϵ > 0 and n large enough, the annealed probability of linear slowdown is bounded from above by exp ( - ( log n ) d - ϵ ) . This bound almost matches the known lower bound of exp ( - C ( log n ) d ) , and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability...

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

Annales de l'I.H.P. Probabilités et statistiques

This paper is concerned with the small time behaviour of a Lévy process X . In particular, we investigate thestabilitiesof the times, T ¯ b ( r ) and T b * ( r ) , at which X , started with X 0 = 0 , first leaves the space-time regions { ( t , y ) ∈ ℝ 2 : y ≤ r t b , t ≥ 0 } (one-sided exit), or { ( t , y ) ∈ ℝ 2 : | y | ≤ r t b , t ≥ 0 } (two-sided exit), 0 ≤ b l t ; 1 , as r ↓ 0 . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L p . In many instances these are...

Small ball probabilities for stable convolutions

Frank Aurzada, Thomas Simon (2007)

ESAIM: Probability and Statistics

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function f : ] 0 , + ∞ [ → ℝ with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as...

Small ball probability estimates in terms of width

Rafał Latała, Krzysztof Oleszkiewicz (2005)

Studia Mathematica

A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have γ ₙ ( s K ) ≤ ( 2 s ) w ² / 4 γ ₙ ( K ) for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

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