We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of $\mathrm {SL}_d({\mathbb {R}})$. As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.

Detecting time-varying target in non-stationary background is difficult and attractive problem. Time-varying movement exists widely in radar and communication systems. The non-linear processing with higher moments is discussed in the situation. Firstly the signal model of time-varying target with fix acceleration is analyzed. Then the radar echoes from synthetic aperture radar (SAR) are processed with higher moments. It not only restrains Gaussian noise automatically, but also suppresses non-stationary noise. Time-varying targets are different from the non-stationary background clutters. Moreover, the influences of the shadows about time-varying targets are reduced in the algorithm of higher moments. The proposal mentioned above utilizes higher moments to avoid analyzing the complex electromagnetic wave propagation and scatter theory. The differences of detection probability are compared between the chip with time-varying target and the clutter chip. The performances among different number order moments are compared by processing lots of actual SAR data added non-stationary noise. Lastly, the suitable number order moments are suggested by comparing the results from processing the actual radar echoes.

A random spatial coverage process whose generating point process is homogeneous Poisson, and whose attached random sets are independent and identically distributed, is called a Boolean model. Motivated by Błaszczyszyn et al. [1], distributional and higher moment properties of the size of clumps (connected clusters of overlapping sets) in this model are derived. This provides some complements to the result on the finiteness of the first moment presented in Hall [3]. The key idea is to construct a certain coupling process for a multitype branching process that dominates the clump size.

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.