Search ResultsShowing 1-3 of 3
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arXiv:1509.05199 (Published 2015-09-17)
Singularity analysis for heavy-tailed random variables
Comments: 23 pages, 4 figuresLet $\alpha \in (0,1),c>0$ and $G(z) = c\sum_{k=1}^\infty z^k \exp( - k^\alpha)$. We apply complex analysis to determine the asymptotic behavior of the $m$th coefficient of the $n$th power of $G(z)$ when $m,n\to \infty$ with $m \geq n G'(1)/G(1)$ and recover five theorems by A. V. Nagaev (1968) on sums of i.i.d. random variables with stretched exponential law $\mathbb{P}(X_1=k) = c \exp(- k ^\alpha)$. From the point of view of complex analysis, the main novelty is the combination of singularity analysis, Lindel\"of integrals, and bivariate saddle points. From the point of view of probability, the proof provides a new, unified approach to large and moderate deviations for heavy-tailed, integer-valued random variables.
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Random partitions in statistical mechanics
Comments: 38 pages, 3 figuresJournal: Electron. J. Probab. 19, 1-37 (2014)DOI: 10.1214/EJP.v19-3244Keywords: statistical mechanics, ideal bose gas, distribution, spatial random partitions, zero-range processTags: journal articleWe consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.
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Cycle structure of random permutations with cycle weights
Comments: 22 pages, 2 figuresJournal: Random Structures & Algorithms 44, 109-133 (2014)DOI: 10.1002/rsa.20430Subjects: 60K35Tags: journal articleWe investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.