We consider a fractional counting process with jumps of amplitude 1, 2, . . . , k, with k ∈ N, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

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 negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards, with probability 1 − α. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.

We consider a semi-Markovian generalization of the integrated telegraph process subject \ud to jumps. It describes a motion on the real line characterized by two alternating velocities \ud with opposite directions, where a jump along the alternating direction occurs at each \ud velocity reversal. We obtain the formal expressions of the forward and backward transition \ud densities of the motion. We express them as series in the case of Erlang-distributed random \ud times separating consecutive jumps. Furthermore, a closed form of the transition density \ud is given for exponentially distributed times, with constant jumps and random initial velocity. \ud In this case we also provide mean and variance of the process, and study the limiting \ud behaviour of the probability law, which leads to a mixture of three Gaussian densities

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the 'expected reversed proportional shortfall order', and a new characterization of random lifetimes involving the reversed hazard rate function.Short title: A quantile-based probabilistic mean value theorem.

We investigate some large deviation problems for a random walk in continuous time {N (t); t ≥ 0} with spatially inhomogeneous rates of alternating type. We first deal with the large deviation principle for the convergence of N (t)/t to a suitable constant. Then, the case of moderate deviations is also discussed. Motivated by possible applications in chemical physics context, we finally obtain an asymptotic lower bound for level crossing probabilities both in the case of finite and infinite horizon.

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U , V , and W are absolutely continuous. The velocities are v = +1 upwards, v = −1 downwards, and v = 0 during idle periods. Let Y + (t), Y − (t), and Y 0 (t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y + (t) is derived. We also obtain the probability law of X(t) = Y + (t) − Y − (t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).