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We introduce and study the cumulative information generating function, which provides a unifying mathematical tool suitable to deal with classical and fractional entropies based on the cumulative distribution function and on the survival function. Specifically, after establishing its main properties and some bounds, we show that it is a variability measure itself that extends the Gini mean semi-difference. We also provide (i) an extension of such a measure, based on distortion functions, and (ii) a weighted version based on a mixture distribution. Furthermore, we explore some connections with the reliability of k-out-of-n systems and with stress–strength models for multi-component systems. Also, we address the problem of extending the cumulative information generating function to higher dimensions. [ABSTRACT FROM AUTHOR]

We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$. The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion. Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D). [ABSTRACT FROM AUTHOR]

We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals Φ (⋅ , ⋅) of two independent squared telegraph processes, both in the case Φ (u , v) = u + v and Φ (u , v) = u ⋅ v. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph process and of the product of two independent symmetric telegraph processes. Some of the expressions obtained are given in terms of new results about derivatives of hypergeometric functions with respect to parameters. [ABSTRACT FROM AUTHOR]