English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we present the distribution of the maximum of the asymmetrictelegraph process in an arbitrary time interval $[0,t]$ under the conditionsthat the initial velocity $V(0)$ is either $c_1$ or $-c_2$ and the number ofchanges of direction is odd or even. For the case $V(0) = -c_2$ the singularcomponent of the distribution of the maximum displays an unexpected cyclicbehavior and depends only on $c_1$ and $c_2$, but not on the current time $t$.We obtain also the unconditional distribution of the maximum for either $V(0) =c_1$ or $V(0) = -c_2$ and its expression has the form of series of Besselfunctions. We also show that all the conditional distributions emerging in thisanalysis are governed by generalized Euler-Poisson-Darboux equations. Werecover all the distributions of the maximum of the symmetric telegraph processas particular cases of the present paper. We underline that it rarely happensto obtain explicitly the distribution of the maximum of a process. For thisreason the results on the range of oscillations of a natural process like thetelegraph model make it useful for many applications.

On the Exact Distributions of the Maximum of the Asymmetric Telegraph Process