Central limit theorems for point processes

Central Limit Theorems For Superlinear Processes

The Central Limit Theorem is studied for stationary sequences that are sums of countable collections of linear processes. Two sets of sufficient conditions are obtained. One restricts only the coefficients and is shown to be best possible among such conditions. The other involves an interplay between the coefficients and the distribution functions of the innovations and is shown to be necessary...

Central Limit Theorems For Dependent, Heterogeneous Processes With Trending Moments

Limit theorems for geometric functionals of Gibbs point processes

Observations are made on a point process Ξ in R in a window Qλ of volume λ. The observation, or ‘score’ at a point x, here denoted ξ(x, Ξ), is a function of the points within a random distance of x. When the input Ξ is a Poisson or binomial point process, the large λ limit theory for the total score ∑ x∈Ξ∩Qλ ξ(x, Ξ ∩Qλ), when properly scaled and centered, is well understood. In this paper we es...

Central Limit Theorems for Arrays of Decimated Linear Processes

Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then establish central limit theorems for arrays of squares of such decimated processes. These theorems are used to obtain the asymptotic behavior of estimators of the s...

Limit Theorems for Renewal Processes

This article describes the Key renewal theorem and the Blackwell’s renewaltheorem. These two limit theorems for renewal processes are equivalent but of different forms.They are particularly useful for characterizing the asymptotic behavior of a probabilisticquantity of interest in a renewal process. We present two applications of these limit theorems:the limiting distributions o...