| 1. |
- Abramowicz, Konrad, et al.
(författare)
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Piecewise multilinear interpolation of a random field
- 2013
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 45:4, s. 945-959
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Tidskriftsartikel (refereegranskat)abstract
- We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.
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| 2. |
- Ahlberg, Daniel, 1982
(författare)
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Asymptotics of First-Passage Percolation on One-Dimensional Graphs
- 2015
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 47:1, s. 182-209
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the Z x {0, 1, ..., K - 1}(d-1) nearest neighbour graph for d, K >= 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0-1 law.
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| 3. |
- Ahlberg, Daniel, et al.
(författare)
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Gilbert's Disc Model With Geostatistical Marking
- 2018
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 50:4, s. 1075-1094
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Tidskriftsartikel (refereegranskat)abstract
- We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in R-2 with radii determined by an underlying stationary and ergodic random field phi: R-2 -> [0, infinity), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of R(2 )does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.
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| 4. |
- Axelson-Fisk, Marina, 1972, et al.
(författare)
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Conditional percolation on one-dimensional lattices
- 2009
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 41:4, s. 3395-3415
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Tidskriftsartikel (refereegranskat)abstract
- Conditioning i.i.d.\ bond percolation with retention parameter $p$ on a one-dimensional periodic lattice on the event of having a bi-infinite path from $-\infty$ to $\infty$ is shown to make sense, and the resulting model exhibits a Markovian structure that facilitates its analysis. Stochastic monotonicity in $p$ turns out to fail in general for this model, but a weaker monotonicity property does hold: the average edge density is increasing in $p$.
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| 5. |
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| 6. |
- Bartoszek, Krzysztof, et al.
(författare)
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NORMAL APPROXIMATION FOR MIXTURES OF NORMAL DISTRIBUTIONS AND THE EVOLUTION OF PHENOTYPIC TRAITS
- 2021
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Ingår i: Advances in Applied Probability. - : APPLIED PROBABILITY TRUST. - 0001-8678 .- 1475-6064. ; 53:1, s. 162-188
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Tidskriftsartikel (refereegranskat)abstract
- Explicit bounds are given for the Kolmogorov andWasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some sigma-algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the sigma-algebra. The proof is based on Steins method. As an application, we consider the Yule-Ornstein-Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.
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| 7. |
- Berzunza Ojeda, Gabriel
(författare)
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The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees
- 2020
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Ingår i: Advances in Applied Probability. - : APPLIED PROBABILITY TRUST. - 0001-8678 .- 1475-6064. ; 52:1, s. 266-290
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider random trees associated with the genealogy of Crump-Mode-Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump-Mode-Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-(2l + 1) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.
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| 8. |
- Bierkens, Joris, et al.
(författare)
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Simulation of elliptic and hypo-elliptic conditional diffusions
- 2020
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 52:1, s. 173-212
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Tidskriftsartikel (refereegranskat)abstract
- Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time 0$ ]]> only linear combinations of its components are observed. That is, one only observes the vector for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.
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| 9. |
- Britton, Tom, et al.
(författare)
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Graphs with specified degree distributions, simple epidemics and local vaccination strategies
- 2007
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 39:4, s. 922-948
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Tidskriftsartikel (refereegranskat)abstract
- Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R-0, the basic reproduction number, and tau(0), the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R-v, the reproduction number, and tau(v), the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.
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| 10. |
- Britton, Tom, et al.
(författare)
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Who is the infector? General multi-type epidemics and real-time susceptibility processes
- 2019
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 51:2, s. 606-631
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Tidskriftsartikel (refereegranskat)abstract
- We couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.
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