伽玛分量驱动的电报过程的一些结果

Pub Date : 2022-06-14 DOI:10.1017/apr.2021.54
B. Martinucci, Alessandra Meoli, S. Zacks
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引用次数: 2

摘要

摘要我们研究了在连续方向反转之间随机时间的一般分布假设下的综合电报过程$X_t$。具体而言,$X_t$表示粒子在时间t处的位置,该粒子以速度c向上移动U个时间单位,以速度$-c$向下移动D个时间单位。根据独立的交替更新,后一种运动循环重复。在以下情况下给出了$X_t$概率律的显式表达式:(i)(U,D)γ分布;(ii)U指数分布和D伽玛分布。对于所涉及参数的某些值,以闭形式给出了$X_t$的概率律,并得到了$X_t的矩母函数及其拉普拉斯变换的一些表达式。后者使我们能够证明Kac型条件的存在,在该条件下,具有相同分布伽马潮间带的集成电报过程的概率密度函数收敛于标准布朗运动的概率密度。最后,我们考虑$X_t$的平方,并揭示了它的分布函数,指定了(U,D)分布的一些选择的表达式。
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Some results on the telegraph process driven by gamma components
Abstract 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).
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