The height of two types of generalised Motzkin paths

Consider Motzkin paths which are lattice paths in the plane starting at the origin, running weakly above the x-axis and after n   unit steps returning at the point (n,0)$(n,0)$. The allowed steps are the up and down steps (1,1)$(1,1)$ and (1,−1)$(1,-1)$ respectively and certain horizontal steps. We consider two types of horizontal steps that have attracted recent attention in the literature. First, we consider unit horizontal steps (1,0)$(1,0)$ coloured with k colours, secondly, we look at paths where the horizontal steps are of length k, for a non-negative integer k. Using generating functions, we study the sum of heights of such paths of size n. With the use of the Mellin transform, we find asymptotic expressions for the mean heights as n tends to infinity.     

L1-Tightness and the law of large numbers in D(IR)

This article presents laws of large numbers for random elements of D(IR). The proofs bring forth the similarities between processes restricted to compact sets of D (IR) and processes with increasing trajectories. The results appear to generalize existing results and cover the independent identically distributed case studied by Ranga Rao.     

Generalized Autoregressive (GAR) Model: A Comparison of Maximum Likelihood and Whittle Estimation Procedures Using a Simulation Study

This article evaluates the performance of two estimators namely, the Maximum Likelihood Estimator (MLE) and Whittle's Estimator (WE), through a simulation study for the Generalised Autoregressive (GAR) model.

As expected, it is found that for the parameters α and σ², the MLE and WE have a better performance than Method of Moments (MOM) estimator. For the parameter δ, MOM sometimes appears to have a slightly better performance than MLE and WE, possibly due to truncation approximations associated with the hypergeometric functions for calculating the autocorrelation function. However, the MLE and WE can be used in practice without loss of efficiency.     

On the statistical inference of a machine-generated autoregressive AR(1) model

We have obtained the asymptotic bias and the limiting distribution for the Yule–Walker estimator of the autoregressive parameter under a considerably weaker assumption than that of independence in the noise sequence. Among other things, these suggest robustness of the classical results and throw some light on the use of simulations based on pseudorandom numbers in verifying these results.