Том 71
№ 6

All Issues

Rodriguez-Dagnino R. M.

Articles: 2
Article (English)

The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

Kolomiets T., Pogorui A. О., Rodriguez-Dagnino R. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 882-889

It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process.

Brief Communications (English)

Multidimensional random motion with uniformly distributed changes of direction and Erlang steps

Pogorui A. О., Rodriguez-Dagnino R. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 572-577

In this paper we study transport processes in $\mathbb{R}^n,\quad n \geq 1$, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we avoid many of the difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang distributed sojourn times.