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Scaling limits of anisotropic Hastings–Levitov clusters

Fredrik Johansson Viklund, Alan Sola, Amanda Turner (2012)

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

We consider a variation of the standard Hastings–Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations...

Semi-recorrido condicionado (expresión asintótica de la r-esperanza condicionada).

Juan Antonio Cuesta Albertos, Carlos Matrán Bea (1983)

Trabajos de Estadística e Investigación Operativa

In a probability space (Ω,σ,P), for α ⊂ σ a sub-σ field, in general the best approximation in L∞ by elements of L∞(α) has not a unique solution. For the election between these, we prove the convergence P-almost surely of the conditional r-means, when r → ∞, to one solution, which we call conditional mid-range. This is characterized for each ω ∈ Ω by the mid-range, of one regular conditional distribution Q(ω, ·).

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 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 deviations of iterated processes in the space of trajectories

Andrei Frolov (2013)

Open Mathematics

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.

Spontaneous clustering in theoretical and some empirical stationary processes*

T. Downarowicz, Y. Lacroix, D. Léandri (2010)

ESAIM: Probability and Statistics

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper...

Statistical convergence of a sequence of random variables and limit theorems

Sanjoy Ghosal (2013)

Applications of Mathematics

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order r and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

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