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Xia A, Yukich JE. Normal approximation for statistics of Gibbsian input in geometric probability. ADV APPL PROBAB 2016. [DOI: 10.1239/aap/1449859795] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This paper concerns the asymptotic behavior of a random variable Wλ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Qλ ↑ ℝd. We establish conditions ensuring that Wλ has volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation for Wλ as λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish a similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.
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Normal approximation for statistics of Gibbsian input in geometric probability. ADV APPL PROBAB 2016. [DOI: 10.1017/s0001867800048965] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
This paper concerns the asymptotic behavior of a random variableWλresulting from the summation of the functionals of a Gibbsian spatial point process over windowsQλ↑ℝd. We establish conditions ensuring thatWλhas volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation forWλas λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish a similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.
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Baryshnikov Y, Eichelsbacher P, Schreiber T, Yukich JE. Moderate deviations for some point measures in geometric probability. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2008. [DOI: 10.1214/07-aihp137] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Schreiber T, Yukich JE. Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. ANN PROBAB 2008. [DOI: 10.1214/009117907000000259] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Heinrich L, Schmidt H, Schmidt V. Limit theorems for functionals on the facets of stationary random tessellations. BERNOULLI 2007. [DOI: 10.3150/07-bej6131] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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