The domain of partial attraction of a Poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean λ of the attracting distribution. In the present paper we start out from results of the recent “probabilistic approach” and derive characterizations for any fixed λ>0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving λ. It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether λ>1 or λ<1. A concrete construction, illustrating all the results, also shows that in the boundary case when λ=1 both possibilities may occur.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45254/1/10959_2005_Article_BF01047000.pd

Characterization of the geometric and exponential random variables

Let the random variable X be distributed over the non-negative integers and let Lm and Rm be the quotient and the remainder in the division of X by m. It is shown that X is geometric if and only if Lm and Rm are independent for m=2,3, . . . . In similar terms is also characterized the exponential random variable

The duals of MMD codes are proper for error detection

A linear code, when used for error detection on a symmetric channel, is said to be proper if the corresponding undetectederror probability increases monotonously in the symbolerror probability of the channel. Such codes are generally considered to perform well in error detection. A number of well-known classes oflinear codes are proper, e.g., the perfect codes, MDS codes, MacDonald\u27s codes, MMDcodes, and some Near-MDS codes. The aim of this work is to show that also the duals of MMD codes are proper

Extended binomial moments of a linear code and the undetected error probability

The extended binomial moments of a linear code, introduced in this paper, are synonymously related to the code weight distribution and linearly to its binomial moments.In contrast to the latter, the extended binomial moments are monotone, which makes them very appropriate for study of the undetected error probability. In this work we establish some properties of the extended binomial moments and based on this we derive new lower and upper bounds on the probability of undetected error. Also, we give a simplification of some previously obtained sufficient conditions for proper and good codes, stated in terms of the extended binomial moments