We consider an iterated function system (with probabilities) of isometries on an unbounded metric space (X, d). Under suitable conditions it is proved that the random orbit {Z n } n 0 of the iterations corresponding to an initial point Z 0 ∈ X 'escapes to infinity' in the sense that P (Z n ∈ K) → 0, as n → ∞ for every bounded set K ⊂ X. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.
We consider the statistical behaviour of independent identically distributed compositions of a finite set of Euclidean isometries of R n . We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalize immediately to Markov chains.We also give simple geometric criteria for orbits to grow linearly or sublinearly with probability one and for nondegeneracy (nonsingular covariance matrix) in the statistical limit theorems.Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.
This study investigates the spatial distribution of ambulance/emergency alarm call events in order to identify spatial covariates associated with the events and discern hotspot regions of the events. The study, which focuses on the Swedish municipality of Skellefteå, is motivated by the problem of developing optimal dispatching strategies for prehospital resources such as ambulances. The dataset at hand is a large-scale multivariate spatial point pattern of call events stretching between the years 2014-2018. For each event, we have recordings of the spatial location of the call as well as marks containing the associated priority level, given by 1 (highest priority) or 2, and sex labels, given by female or male. To achieve our goals, we begin by modeling the spatially varying call occurrence risk as an intensity function of a (multivariate) inhomogeneous spatial Poisson process that we assume is a log-linear function of some underlying spatial covariates. The spatial covariates used in this study are related to road network coverage, population density, and the socio-economic status of the population in Skellefteå. Since mobility is clearly a factor that has a large impact on where people are in need of an ambulance, and since none of our spatial covariates quantify human mobility patterns, we here take a pragmatic approach where, in addition to other spatial covariates, we include a non-parametric intensity estimate of the events as a covariate in the intensity function. A new heuristic algorithm has been developed to select an optimal estimate of the kernel bandwidth in order to obtain the non-parametric intensity estimate of the events and to generate other covariates. Since we consider a large number of spatial covariates as well as their products (the first-order interaction terms), and since some of them may be strongly correlated, lasso-like elastic-net regularisation has been used in the log-likelihood intensity modeling to perform variable selection and reduce variance inflation from overfitting and bias from underfitting. As a result of the variable selection, the fitted model structure contains individual covariates of both road network and demographic types. We discovered that hotspot regions of calls have been observed along dense parts of the road network in Skellefteå. Furthermore, a mean absolute error evaluation of the proposed model to generate the intensity of emergency alarm/ambulance call events indicates that the estimated model is stable and can be used to generate a reliable intensity estimate over the region, which can be used as an input in the problem of designing prehospital resource dispatching strategies.
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