1,167 research outputs found
Convergence of Markov processes near saddle fixed points.
We consider sequences (XtN)tā„0 of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form xĢt=b(xt), where for some Ī», Ī¼>0 and Ļ(x)=O(|x|2). Here the processes are indexed so that the variance of the fluctuations of XtN is inversely proportional to N. The simplest example arises from the OK Corral gunfight model which was formulated by Williams and McIlroy [Bull. London Math. Soc. 30 (1998) 166ā170] and studied by Kingman [Bull. London Math. Soc. 31 (1999) 601ā606]. These processes exhibit their most interesting behavior at times of order logN so it is necessary to establish a fluid limit that is valid for large times. We find that this limit is inherently random and obtain its distribution. Using this, it is possible to derive scaling limits for the points where these processes hit straight lines through the origin, and the minimum distance from the origin that they can attain. The power of N that gives the appropriate scaling is surprising. For example if T is the time that XtN first hits one of the lines y=x or y=āx, then NĪ¼/{(2(Ī»+Ī¼))}|XTN| ā |Z|Ī¼/{(Ī»+Ī¼)}, for some zero mean Gaussian random variable Z
Weak convergence of the localized disturbance flow to the coalescing Brownian flow
We define a new state-space for the coalescing Brownian flow, also known as
the Brownian web, on the circle. The elements of this space are families of
order-preserving maps of the circle, depending continuously on two time
parameters and having a certain weak flow property. The space is equipped with
a complete separable metric. A larger state-space, allowing jumps in time, is
also introduced, and equipped with a Skorokhod-type metric, also complete and
separable. We prove that the coalescing Brownian flow is the weak limit in this
larger space of a family of flows which evolve by jumps, each jump arising from
a small localized disturbance of the circle. A local version of this result is
also obtained, in which the weak limit law is that of the coalescing Brownian
flow on the line. Our set-up is well adapted to time-reversal and our weak
limit result provides a new proof of time-reversibility of the coalescing
Brownian flow. We also identify a martingale associated with the coalescing
Brownian flow on the circle and use this to make a direct calculation of the
Laplace transform of the time to complete coalescence.Comment: Published at http://dx.doi.org/10.1214/13-AOP845 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: substantial text
overlap with arXiv:0810.021
Partial stochastic dominance for the multivariate Gaussian distribution
Gaussian comparison inequalities provide a way of bounding probabilities
relating to multivariate Gaussian random vectors in terms of probabilities of
random variables with simpler correlation structures. In this paper, we
establish the partial stochastic dominance result that the cumulative
distribution function of the maximum of a multivariate normal random vector,
with positive intraclass correlation coefficient, intersects the cumulative
distribution function of a standard normal random variable at most once. This
result can be applied to the Bayesian design of a clinical trial in which
several experimental treatments are compared to a single control.Comment: 7 page
Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure
Scenario generation is the construction of a discrete random vector to
represent parameters of uncertain values in a stochastic program. Most
approaches to scenario generation are distribution-driven, that is, they
attempt to construct a random vector which captures well in a probabilistic
sense the uncertainty. On the other hand, a problem-driven approach may be able
to exploit the structure of a problem to provide a more concise representation
of the uncertainty.
In this paper we propose an analytic approach to problem-driven scenario
generation. This approach applies to stochastic programs where a tail risk
measure, such as conditional value-at-risk, is applied to a loss function.
Since tail risk measures only depend on the upper tail of a distribution,
standard methods of scenario generation, which typically spread their scenarios
evenly across the support of the random vector, struggle to adequately
represent tail risk. Our scenario generation approach works by targeting the
construction of scenarios in areas of the distribution corresponding to the
tails of the loss distributions. We provide conditions under which our approach
is consistent with sampling, and as proof-of-concept demonstrate how our
approach could be applied to two classes of problem, namely network design and
portfolio selection. Numerical tests on the portfolio selection problem
demonstrate that our approach yields better and more stable solutions compared
to standard Monte Carlo sampling
Planar aggregation and the Brownian web
Diffusion limited aggregation (DLA) is a random growth model which was originally introduced in 1981 by Witten and Sander. This model is prevalent in nature and has many applications in the physical sciences as well as industrial processes. We consider a simplified version of DLA known as the Hastings-Levitov HL(0) model, and show that under certain scaling conditions this model gives rise to a limit object known as the Brownian web
Is Clean Air Possible? A Critical Analysis of China's Intended Nationally Determined Contributions for the United Nations Climate Change Conference 2015.
In December 2015 at the 21st Conference of the Parties of the United Nations Framework Convention on Climate Change in Paris, 195 countries signed to the world's first comprehensive climate change agreement; The Paris Agreement. Subsequently 155 of these countries have ratified the agreement. This is makes apparent the global consensus that global climate change is fast becoming the most important issue facing our world today. Being the largest contributor to global emission, China is a significant player in this agreement and therefore their Intended Nationally Determined Contributions merit a closer examination. Under the SWOT framework, China's Intended Nationally Determined Contributions have been subjected to a critical policy analysis, a comparative analysis, and a feasibility study. From this, there is a determination of the likelihood of implementation. While it is determined that the Intended Nationally Determined Contributions are relatively feasible for China, there are complications in relation to the domestic enforceability of the policies that would result from changes made to current policy. The National Government recognizes the need for more enforceability of policies, however, it remains questionable as to whether coherent and cooperative enforcement will be achievable
Scaling limits and fluctuations for random growth under capacity rescaling
We evaluate a strongly regularised version of the Hastings-Levitov model HL for . Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on . Furthermore, this field becomes degenerate as approaches 0 and 2, suggesting the existence of phase transitions at these values
Scaling limits of anisotropic growth on logarithmic time-scales
We study the anisotropic version of the Hastings-Levitov model AHL.
Previous results have shown that on bounded time-scales the harmonic measure on
the boundary of the cluster converges, in the small-particle limit, to the
solution of a deterministic ordinary differential equation. We consider the
evolution of the harmonic measure on time-scales which grow logarithmically as
the particle size converges to zero and show that, over this time-scale, the
leading order behaviour of the harmonic measure becomes random. Specifically,
we show that there exists a critical logarithmic time window in which the
harmonic measure flow, started from the unstable fixed point, moves
stochastically from the unstable point towards a stable fixed point, and we
show that the full trajectory can be characterised in terms of a single
Gaussian random variable.Comment: 29 pages, 5 figure
The diagenesis and palaeomagnetism of permian and triassic sediments from central Spain
Detailed diagenetic and palaeomagnetic studies have been made of Permian and Triassic rocks from the Iberian Cordillera, Spain. Five stratigraphical units comprising the Autunian, Saxonian, Buntsandstein, Muschelkalk, and Keuper have been studied in a number of sections which have been well documented sedimentologically. Autuninan rocks have a characteristic remanence which is exclusively reversed and corresponds to the Kiaman Interval. The pole position identified is consistent with previous studies, which indicate the rotation of Iberia in post-Triassic times. The Saxonian facies show complex multicomponent magnetizations; no polarity zonation can be resolved. The Buntsandstein is remarkable in that much of it is remagnetised in a direction similar to the present-day local geomagnetic field direction. The secondary remanence is carried by fine-grained haematite which has been formed as a result of carbonate dissolution associated with structural inversion of the Iberian Cordillera. Dating of diagenetic events associated with this remagnetization is also possible. Fragments of primary remanence are preserved in some fine-grained lithologies of the Buntsandstein. These indicate that normal and reversed zones of magnetization were originally present. The magnetization of the Muschelkalk and Keuper carbonates is also complex; secondary components similar to those of the Buntsandstein are present but there is evidence that primary components were predominantly normal during Karnian times
A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order RungeāKutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method
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