Asymptotic properties of near Pfeifer records

Asymptotic properties of the number of near records are known in the literature. We generalize these results to the Pfeifer model which has a wider application. In particular we establish convergence in probability, in the almost sure sense and in distribution for the number of near records under the Pfeifer model

Convergence of tail sum for records

Suppose {Rn(L) (F): n≥1} is the sequence of lower records from a distribution F, where F is continuous with inf {x ∈ supp(F)}=0. We derive conditions under which logarithm of the tail sum of records, ∑ j=n ∞=Rn (L) (F), properly centered and scaled, converge weakly. We also prove two results on ?-varying and regularly varying functions, which are of independent interest

Convergence of linear functions of Pfeifer records

The probabilistic behaviour of partial sums of upper and lower records has been studied in the literature. In this article, we take a broader view and study partial sums of record like sequences. We show that such sequences converge in distribution to normal and lognormal distribution. In particular our results apply to Pfeifer records. We also show the strong convergence of partial sums of lower Pfeifer records under suitable assumptions

Limiting spectral distribution of XX ′ matrices

The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) [7] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. Bose and Sen (2008) [6] have developed this method further and have provided a general framework which deals with symmetric matrices with entries coming from an independent sequence. In this article we enlarge the scope of the above approach to consider matrices of the form Ap = 1 n XX ′ where X is a p × n matrix with real entries. We establish some general results on the existence of the spectral distribution of such matrices, appropriately centered and scaled, when p → ∞ and n = n(p) → ∞ and p/n → y with 0 ≤ y < ∞. As examples we show the existence of the spectral distribution when X is taken to be the appropriate asymmetric Hankel, Toeplitz, circulant and reverse circulant matrices. In particular, when y = 0, the limits for all these matrices coincide and is the same as the limit for the symmetric Toeplitz derived in Bryc, Dembo and Jiang (2006) [7]. In other cases, we obtain new limiting spectral distributions for which no closed form expressions are known. We demonstrate the nature of these limits through some simulation results

Partial sum process for records

Suppose the upper records {X <SUB>Ln</SUB>} from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z<SUB>n</SUB>(.)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z<SUB>n</SUB>} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal

Random oriented trees: a model of drainage networks

Consider the d-dimensional lattice Ζd where each vertex is "open" or "closed" with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d)=v(d)-1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d=2 and 3 and it is an infinite collection of distinct trees for d≥4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l