scholarly journals Random Inscribing Polytopes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Ross M. Richardson ◽  
Van H. Vu ◽  
Lei Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Convex Bodies ◽  
Higher Moments ◽  
Random Polytopes ◽  
International Audience

International audience For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

2015 ◽  
Vol 52 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Raúl Fierro ◽  
Víctor Leiva ◽  
Jesper Møller
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Hawkes Process ◽  
Homogeneous Poisson Process ◽  
Large Numbers ◽  
Main Shocks

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

scholarly journals Point process stabilization methods and dimension estimation

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Point Processes ◽  
Dimension Estimation ◽  
Stabilization Methods ◽  
International Audience

International audience We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.

A note on the central limit theorem for doubly stochastic Poisson processes

1976 ◽  
Vol 13 (04) ◽  
pp. 809-813
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
The Central Limit Theorem ◽  
Doubly Stochastic Poisson Process ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

scholarly journals Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

2014 ◽  
Vol 46 (2) ◽  
pp. 348-364 ◽  
Author(s):  
Günter Last ◽  
Mathew D. Penrose ◽  
Matthias Schulte ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Malliavin Calculus ◽  
Limit Theorems ◽  
Measurable Space ◽  
Central Limit Theorems ◽  
Poisson Processes ◽  
Chaos Expansion

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in .

scholarly journals Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

2014 ◽  
Vol 46 (02) ◽  
pp. 348-364 ◽  
Author(s):  
Günter Last ◽  
Mathew D. Penrose ◽  
Matthias Schulte ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Malliavin Calculus ◽  
Limit Theorems ◽  
Measurable Space ◽  
Central Limit Theorems ◽  
Poisson Processes ◽  
Chaos Expansion

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a PoissonU-statistic. The approach is based on recent results of Peccatiet al.(2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process ofk-dimensional flats in.

scholarly journals The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

2015 ◽  
Vol 52 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Raúl Fierro ◽  
Víctor Leiva ◽  
Jesper Møller
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Hawkes Process ◽  
Homogeneous Poisson Process ◽  
Large Numbers ◽  
Main Shocks

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

scholarly journals The degree distribution in unlabelled $2$-connected graph families

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Veronika Kraus
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Connected Graph ◽  
Degree Distribution ◽  
Random Variable ◽  
Expected Value ◽  
Asymptotically Linear ◽  
International Audience ◽  
Graph Families

International audience We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families. We prove a central limit theorem for $X_n^k$ with expected value $\mathbb{E}X_n^k \sim \mu_kn$ and variance $\mathbb{V}X_n^k \sim \sigma_k^2n$, both asymptotically linear in $n$, for both rooted and unrooted unlabelled $2$-connected outerplanar or series-parallel graphs.

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