On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

Word frequencies: A comparison of Pareto type distributions

Mehri and Jamaati (2017) [18] used Zipf's law to model word frequencies in Holy Bible translations for one hundred live languages. We compare the fit of Zipf's law to a number of Pareto type distributions. The latter distributions are shown to provide the best fit, as judged by a number of comparative plots and error measures. The fit of Zipf's law appears generally poor.     

Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences

Let \((X_{n}^{\ast})\) be an independent identically distributed random sequence. Let \(M_{n}^{\ast}\) and \(m_{n}^{\ast}\) denote, respectively, the maximum and minimum of \(\{X_{1}^{\ast},\cdots,X_{n}^{\ast}\}\). Suppose that some of the random variables \(X_1^{\ast},X_2^{\ast},\cdots\) can be observed and let \(\widetilde{M}_n^{\ast}\) and \(\widetilde{m}_n^{\ast}\) denote, respectively, the maximum and minimum of the observed random variables from the set \(\{X_1^{\ast},\cdots,X_n^{\ast}\}\). In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector \((\widetilde{M}_n^{\ast}, \widetilde{m}_n^{\ast}, M_n^{\ast}, m_n^{\ast})\). The results are extended to weakly dependent stationary Gaussian sequences.     

Limit distributions for the bivariate geometric maxima

Limit laws are established for the behavior of (max X _i , max Y _i ) when (X _i , Y _i ) are independent and distributed according to a bivariate geometric distribution.     

The Distribution of the Bit Error Rate for an M Branch Antenna with N Interferers

Given an antenna with M branches, the bit error rate (BER) and mean squared error (MSE) for choosing the antenna weights (to approximately cancel M???1 interferers), are given by $$ \mathit{BER} \approx C \;\exp \left(-\alpha-\alpha Z_N\right) \mbox{ and } \mathit{MSE}=1/\left(1+Z_N\right), $$ where Z _N is the signal-to-interference plus noise ratio and C, α are some fixed parameters. So, obtaining the distribution of Z _N allows one to obtain the distribution of the MSE and to approximate that of the BER. Three cases are presented:

• the case of fixed powers for the interferers, say Q ₁, ..., Q _N , and for the wanted signal, say Q ₀;
• the case of fixed power for the wanted signal and random powers for the interferers;
• the case of random powers for both the wanted signal and the interferers.

We assume that Q ₀,...,Q _N are independent with different distributions. We show that to magnitude 1/N, the distribution of Z is just that of Q ₀ g _M /T, where g _M is a gamma random variable with mean M and T is the average of the total interferer power: $$ T = \mathbb{E} \ \left\{ \sum\limits_{j=1}^N Q_j\right\}. $$ We also show how to obtain an expansion in powers of 1/N for the distribution of $\mathit{TZ}$ about that of Q ₀ g _M . For example, to get the distribution of $\mathit{TZ}$ up to magnitude 1/N ², one requires only the means of Q ₁,...,Q _N and $Q_1^2,\ldots,Q_N^2$ and the distribution of Q ₀.     

On modified Chebyshev polynomials

Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321-343].