995 research outputs found

    Critical scaling of stochastic epidemic models

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    In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent pp-coin tosses. Spatial variants of these models are proposed, in which finite populations of size NN are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter pp is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift.Comment: Published at http://dx.doi.org/10.1214/074921707000000346 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

    Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

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    Let Υ\Upsilon be a compact, negatively curved surface. From the (finite) set of all closed geodesics on Υ\Upsilon of length L\leq L, choose one, say γL\gamma_{L}, at random and let N(γL)N (\gamma_{L}) be the number of its self-intersections. It is known that there is a positive constant κ\kappa depending on the metric such that N(γL)/L2κN (\gamma_{L})/L^{2} \rightarrow \kappa in probability as LL\rightarrow \infty. The main results of this paper concern the size of typical fluctuations of N(γL)N (\gamma_{L}) about κL2\kappa L^{2}. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order LL, in particular, (N(γL)κL2)/L(N (\gamma_{L})-\kappa L^{2})/L converges weakly to a nondegenerate probability distribution. In contrast, it is also proved that if the metric has variable negative curvature then fluctuations of N(γL)N (\gamma_{L}) are of order L3/2L^{3/2}, in particular, (N(γL)κL2)/L3/2(N (\gamma_{L})-\kappa L^{2})/L^{3/2} converges weakly to a Gaussian distribution. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure

    Spatial Epidemics: Critical Behavior in One Dimension

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    In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are proposed, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson-Watanabe processes with killing, respectively

    Consistency of Bayes estimators of a binary regression function

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    When do nonparametric Bayesian procedures ``overfit''? To shed light on this question, we consider a binary regression problem in detail and establish frequentist consistency for a certain class of Bayes procedures based on hierarchical priors, called uniform mixture priors. These are defined as follows: let ν\nu be any probability distribution on the nonnegative integers. To sample a function ff from the prior πν\pi^{\nu}, first sample mm from ν\nu and then sample ff uniformly from the set of step functions from [0,1][0,1] into [0,1][0,1] that have exactly mm jumps (i.e., sample all mm jump locations and m+1m+1 function values independently and uniformly). The main result states that if a data-stream is generated according to any fixed, measurable binary-regression function f0≢1/2f_0\not\equiv1/2, then frequentist consistency obtains: that is, for any ν\nu with infinite support, the posterior of πν\pi^{\nu} concentrates on any L1L^1 neighborhood of f0f_0. Solution of an associated large-deviations problem is central to the consistency proof.Comment: Published at http://dx.doi.org/10.1214/009053606000000236 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions

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    Consider a critical nearest neighbor branching random walk on the dd-dimensional integer lattice initiated by a single particle at the origin. Let GnG_{n} be the event that the branching random walk survives to generation nn. We obtain limit theorems conditional on the event GnG_{n} for a variety of occupation statistics: (1) Let VnV_{n} be the maximal number of particles at a single site at time nn. If the offspring distribution has finite α\alphath moment for some integer α2\alpha\geq 2, then in dimensions 3 and higher, Vn=Op(n1/α)V_n=O_p(n^{1/\alpha}); and if the offspring distribution has an exponentially decaying tail, then Vn=Op(logn)V_n=O_p(\log n) in dimensions 3 and higher, and Vn=Op((logn)2)V_n=O_p((\log n)^2) in dimension 2. Furthermore, if the offspring distribution is non-degenerate then P(VnδlognGn)1P(V_n\geq \delta \log n | G_{n})\to 1 for some δ>0\delta >0. (2) Let Mn(j)M_{n} (j) be the number of multiplicity-jj sites in the nnth generation, that is, sites occupied by exactly jj particles. In dimensions 3 and higher, the random variables Mn(j)/nM_{n} (j)/n converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a "typical" site (that is, at the location of a randomly chosen particle of the nnth generation) is of order Op(logn)O_p(\log n), and the number of occupied sites is Op(n/logn)O_p(n/\log n)
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