Related papers: On the Exact Distributions of the Maximum of the A…

In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive…

We consider the telegraph process with two velocities, $a_1>a_2\in\mathbb{R}$, and two rates of reversal, $\lambda_1,\lambda_2>0$. We study some of its features with respect to the conditional probability measure where both the initial…

In this paper we study the joint distributions of the telegraph process and its maximum conditioned on the number of changes of direction and the initial velocity. We prove that in the case of positive starting velocity, a form of the…

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated to the probability that the process remains below a…

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or…

In this note, we present some ideas for describing the distributions of the running maximum/minimum, first passage times and telegraphic meanders. Explicit formulae for joint distribution of the extrema, the number of velocity switches and…

We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities $ \pm c $ and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study…

We study time series concerning rare events. The occurrence of a rare event is depicted as a jump of constant intensity always occurring in the same direction, thereby generating an asymmetric diffusion process. We consider the case where…

Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$. The processes $X_k(t), \; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \; c_2>0,$ that start at…

We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\rm m}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we…

In this paper we present the distribution of the telegraph meander, a random function obtained by conditioning the telegraph process to stay above the zero level. The reflection principle for finite-velocity random motions allows the law of…

In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed…

Of stochastic differential equations, diffusion processes have been adopted in numerous applications, as more relevant and flexible models. This paper studies diffusion processes in a different setting, where for a given stationary…

We give the distribution function of $M_n$, the maximum of a sequence of $n$ observations from an autoregressive process of order 2. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random…

The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant…

We consider the asymptotic behavior of the expectation of the maximum for a special assignment process with constant or i.i.d. coefficients. We show how it depends on the coefficients' distribution.

A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of…

We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$, both developing with finite constant speed $c>0$, that, at the initial time instant $t=0$, simultaneously start from the origin…

We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities $v_1$ and $v_2$. We assume that the sequences of random times concerning the motions along…

For the one-dimensional telegraph process, we obtain explicit distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of…