Multidimensional Renewal Research Articles

Some properties of multidimensional renewal processes

Multidimensional Poisson renewal processes are considered. Analytical expressions are derived for the main characteristics of the processes and some properties of the multidimensional renewal equations and renewal functions are examined.

A multi-dimensional renewal theorem for finite Markov chains

Let U, L and F be functions from Z d into the set of real square matrices of finite dimension N, and let in addition L(t) be positive for each t. Define the convolution L*U by the formula L*U(t)=Σ t1 + t2 = t L(t 1 )U(t 2 ), and put R=Σ ∞ n=0 L n* , provided the sum converges. Here L 0* =δ, where δ(0)=1 (the identity matrix) and δ(t)=0 for t¬=0, and L n* =L*L (n−1)* for n≥1. A solution U of the renewal equation U-L*U=F is then given by U=R*F, provided the latter expression converges. The object of the paper is to study the asymptotic behaviour of R*F(t), as |t|→∞. The result can be applied to first passage problems for sums of Markov dependent random variables

Strong Approximation of Extended Renewal Processes

We develop a strong approximation approach to extended multidimensional renewal theory. The consequences of this approximation are a Bahadur-Kiefer type representation of the renewal process in terms of partial sums, Strassen and Chung type laws of the iterated logarithm. We also give a characterization of the renewal process by four classes of deterministic curves in the sense of Revesz (1982). We generalize our results to the case of non-independent and/or nonidentically distributed random vectors.

On the multi-dimensional renewal theorem

Let X be a random variable on R n , n ⩾ 2, having a density. Assume X has a finite exponential moment and non-zero mean vector, μ. Let ν be the corresponding renewal measure, and Q a cube. We obtain an asymptotic formula for ν( x + Q) as x → ∞ which is uniform in a small cone about the mean vector. This formula depends on moments of arbitrarily high order but depends only on the first and second moments of X in a region x · μ > ¦x¦¦μ¦(1 − o(¦x¦ −2 3 )) .

A Note on a Multi-Dimensional Renewal Equation

A Note on a Multi-Dimensional Renewal Equation

A Multidimensional Renewal Theorem

A multidimensional renewal theorem is proven which generalizes the well-known result of Pyke on Markov renewal processes. The result is a natural setting of the multidimensional renewal theorem, and has applications to age-dependent branching processes.

Renewal theory in two dimensions: asymptotic results

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Renewal theory in two dimensions: asymptotic results

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Multidimensional Renewal Equations and Moments of Branching Processes

Multidimensional Renewal Equations and Moments of Branching Processes

Infinite Particle Systems and Multi-Dimensional Renewal Theory

Infinite Particle Systems and Multi-Dimensional Renewal Theory