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Register a new userThe divisible sandpile with heavy-tailed variables
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This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an $alpha$-stable random distribution.
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The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize. The associated odometer function describes the amount of mass emitted from each vertex during stabilization. In this survey we describe recent scaling limit results of the odometer function depending on different initial configurations and redistribution rules. Moreover we review connections to the obstacle problem from potential theory, including the connection between odometers and limiting shapes of growth models such as iDLA. Finally we state some open problems.
We study the statistics of the largest eigenvalues of $p times p$ sample covariance matrices $Sigma_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-alpha}$, $alpha>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{mu+1}$, $0 leq mu leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $alpha<2(1+mu^{{-1}})$ and converge to a constant in the case $alpha>2(1+mu^{{-1}})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.
We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<alpha<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.
Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.
In many applications, significant correlations between arrivals of load-generating events make the numerical evaluation of the load of a system a challenging problem. Here, we construct very accurate approximations of the workload distribution of the MAP/G/1 queue that capture the tail behavior of the exact workload distribution and provide a small relative error. Motivated by statistical analysis, we assume that the service times are a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive our approximations as a sum of the workload distribution of the MAP/PH/1 queue and a heavy-tailed component that depends on the perturbation parameter. We refer to our approximations as corrected phase-type approximations, and we exhibit their performance with a numerical study.
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