Hausdorff and packing dimensions of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.

Method for determining the fractional part of the carrier frequency cycle of the simulator GNSS signals

The article presents the calibration problem of a navigation signals simulator while ensuring the measurements uniformity of coordinate and time measuring instruments. The method of determining the carrier frequency cycle fractional part in the simulator RF circuit output and standart deviation estimate of the pseudorange generation error by the simulator on the carrier phase with using of an oscilloscope is presented. The method essence is to determine the phase difference between two signals: the simulator absolute calibration with a single RF output – the phase difference between the navigation signal generated by the simulator and the reference harmonic signal; the simulator relative calibration with two or more RF outputs – the phase difference between the navigation signals generated by the simulator from different RF outputs. The applying of the simulator absolute calibration by the carrier frequency phase to solve the receiver calibration problem by the carrier frequency phase is shown, which will significantly simplify the implementation of the promising Integer-PPP technology. Presented the use of the simulator relative calibration to determine the accuracy characteristics of angular receiver in its development and testing.

Exponential Autoregressive Parameters Estimation

The purpose of the research was to compare the least squares estimatate and the least absolute deviation estimate depending on the probability distribution of the renewal process of the autoregressive equation. To achieve this goal, the sequence of observations of the exponential autoregressive process was repeatedly reproduced using computer simulation, and the least squares estimate and the least absolute deviation estimate were calculated for each sequence. The resulting estimation sequences were used to calculate the sample variances of the least squares estimate and the least absolute deviation estimate. The best estimate was the one with the lowest sample variance. The quantitative measure for the estimates comparison was the sample relative efficiency of estimates, defined as the inverse ratio of their sample variances. Normal distribution, contaminated normal distribution, i.e. Tukey distribution, with different values of the proportion and intensity of contamination, logistic distribution, Laplace distribution and Student distribution with different degrees of freedom, in particular, with one degree of freedom, that is, Cauchy distribution, were used as models of probability distribution of the renewal process. For each probability distribution, asymptotic values of the sample relative efficiency were obtained with an unlimited increase in the sample size of the observations of the autoregressive process. Findings of research show that the least absolute deviation estimate is better than the least squares estimate for Laplace distribution and the contaminated normal distribution with sufficiently large levels of the proportion and intensity of contamination. In other cases, the least squares estimate is preferable.