Displaying similar documents to “Entropy of random walk range”

Quantitative recurrence in two-dimensional extended processes

Françoise Pène, Benoît Saussol (2009)

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

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Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighbourhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including ℤ-extension of mixing subshifts of finite type. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a result of convergence...

Large deviations for voter model occupation times in two dimensions

G. Maillard, T. Mountford (2009)

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

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We study the decay rate of large deviation probabilities of occupation times, up to time , for the voter model : ℤ×[0, ∞)→{0, 1} with simple random walk transition kernel, starting from a Bernoulli product distribution with density ∈(0, 1). In [ (1988) 401–413], Bramson, Cox and Griffeath showed that the decay rate order lies in [log(), log()]. In this paper, we establish the true decay rates depending on the level. We show that the decay rates are log() when the deviation...

Metric entropy of convex hulls in Hilbert spaces

Wenbo Li, Werner Linde (2000)

Studia Mathematica

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Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), T = t 1 , t 2 , . . . , | | t j | | ≤ a j , by functions of the a j ’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences...

Sparsity in penalized empirical risk minimization

Vladimir Koltchinskii (2009)

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

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Let (, ) be a random couple in × with unknown distribution . Let ( , ), …, ( , ) be i.i.d. copies of (, ), being their empirical distribution. Let , …, :↦[−1, 1] be a dictionary consisting of functions. For ∈ℝ, denote :=∑ . Let :×ℝ↦ℝ be a given loss function, which is convex with...