No abstract

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, have attracted much attention. The [Formula: see text]th moment of the limit equals the number of non-crossing pair-partitions of the set [Formula: see text]. There are several extensions of this result in the literature. In this paper, we consider a unifying extension which also yields additional results. Suppose [Formula: see text] is an [Formula: see text] symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems. Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős–Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

In this paper, we consider a generalized kernel smoothing estimator of the regression function with non-negative support, using gamma probability densities as kernels, which are nonnegative and have naturally varying shapes. It is based on a generalization of Hille's lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method.

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, has attracted much attention. The 2kth moment of the limit equals the number of non-crossing pair-partitions of the set {1, 2, . . . , 2k}. There are several extensions of this result in the literature. In this paper we consider a unifying extension which also yields additional results.Suppose W n is an n × n symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems.Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős-Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

We obtain the ruin probability and expected discounted penalty function for a diffusion-perturbed classical risk model, by taking limits in a sequence of compound Poisson processes that converge weakly to the former. This allows us to improve upon a result of Tsai and Willmot [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance Math. Econ. 30, 51-66].