| 1. |
- Kennerberg, Philip, et al.
(författare)
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Jante's law process
- 2018
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 50:2, s. 414-439
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Tidskriftsartikel (refereegranskat)abstract
- Consider the process which starts with N ≥ 3 distinct points on ℝd, and fix a positive integer K < N. Of the total N points keep those N - K which minimize the energy amongst all the possible subsets of size N - K, and then replace the removed points by K independent and identically distributed points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite nonrestrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the `Keynesian beauty contest process' introduced in Grinfeld et al. (2015), where K = 1 and the distribution ζ was uniform on the unit cube.
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| 2. |
- Basak, Gopal, et al.
(författare)
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An urn model for odds ratio based adaptive phase III clinical trials
- 2008
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Ingår i: Markov Processes and Related Fields. - 1024-2953. ; 14:4, s. 571-582
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Tidskriftsartikel (refereegranskat)abstract
- We study the limiting behaviour of a generalized Polya urn, motivated by adaptive data-dependent allocation designs, which are used in Phase III clinical trials in order to allocate a larger number of patients to the better treatment. We establish rigorous limiting results for the model, including the Central Limit Theorem, thus providing the theoretical background for using the odds ratio-based adaptive designs.
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| 3. |
- Basak, Gopal, et al.
(författare)
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Snakes and perturbed random walks
- 2013
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Ingår i: Proceedings of the Steklov Institute of Mathematics. - 0081-5438. ; 282:1, s. 42-51
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Tidskriftsartikel (refereegranskat)abstract
- We study some properties of random walks perturbed at extrema, which are generalizations of the walks considered, e.g., by Davis (1999) and Tth (1996). This process can also be viewed as a version of an excited random walk, recently studied by many authors. We obtain several properties related to the range of the process with infinite memory and prove the strong law, the central limit theorem, and the criterion for the recurrence of the perturbed walk with finite memory. We also state some open problems. Our methods are predominantly combinatorial and do not involve complicated analytic techniques.
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| 4. |
- Comets, Francis, et al.
(författare)
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Generalizations of forest fires with ignition at the origin
- 2023
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 60:2, s. 418-434
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Tidskriftsartikel (refereegranskat)abstract
- We study generalizations of the forest fire model introduced in [4] and [10] by allowing the rates at which the trees grow to depend on their location, introducing long-range burning, as well as a continuous-space generalization of the model. We establish that in all the models in consideration the expected time required to reach a site at distance x from the origin is of order for any 0$ ]]>.
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| 5. |
- Crane, Edward, et al.
(författare)
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The limit point in the Jante's law process has an absolutely continuous distribution
- 2024
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Ingår i: Stochastic Processes and their Applications. - 0304-4149. ; 168
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Tidskriftsartikel (refereegranskat)abstract
- We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante's law process. We consider a version of the model where the space of possible opinions is a convex body B in Rd. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.
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| 6. |
- Crane, Edward, et al.
(författare)
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The simple harmonic urn
- 2011
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Ingår i: Annals of Probability. - 0091-1798. ; 39:6, s. 2119-2177
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Tidskriftsartikel (refereegranskat)abstract
- We study a generalized Pólya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth–death processes, a uniform renewal process, the Eulerian numbers, and Lamperti’s problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a “Poisson earthquakes” Markov chain on the homeomorphisms of the plane.
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| 7. |
- Engländer, János, et al.
(författare)
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Conservative random walk*
- 2022
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 27
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Tidskriftsartikel (refereegranskat)abstract
- Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in Zd, d ≥ 2: at time n the direction of the process is “updated” with probability pn; otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits. Our results complement previous ones in the literature about “correlated” (or “Newtonian”) and “persistent” random walks.
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| 8. |
- Engländer, János, et al.
(författare)
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Impatient Random Walk
- 2019
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Ingår i: Journal of Theoretical Probability. - : Springer Science and Business Media LLC. - 0894-9840 .- 1572-9230. ; 32:4, s. 2020-2043
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new type of random walk where the definition of edge repellence/reinforcement is very different from the one in the “traditional” reinforced random walk models and investigate its basic properties, such as null versus positive recurrence, transience, as well as the speed. The two basic cases will be dubbed “impatient” and “ageing” random walks.
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| 9. |
- Engländer, János, et al.
(författare)
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The coin-turning walk and its scaling limit
- 2020
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 25
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Tidskriftsartikel (refereegranskat)abstract
- Let S be the random walk obtained from “coin turning” with some sequence {pn}n≥2, as introduced in [8]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n−1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.
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| 10. |
- Engländer, János, et al.
(författare)
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Turning a Coin over Instead of Tossing It
- 2018
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Ingår i: Journal of Theoretical Probability. - : Springer Science and Business Media LLC. - 0894-9840 .- 1572-9230. ; 31:2, s. 1097-1118
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Tidskriftsartikel (refereegranskat)abstract
- Given a sequence of numbers (Formula presented.) in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability (Formula presented.), (Formula presented.), independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as (Formula presented.)? We show that a number of phase transitions take place as the turning gets slower (i. e., (Formula presented.) is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is (Formula presented.). Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.
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