Recent studies show that stable distributions are successful in modeling heavy-tailed or impulsive noise. Investigation of the stability of a probability distribution can be greatly facilitated if the corresponding characteristic function (CF) has a closed-form expression. We explore a new family of distribution called the Vertically-Drifted First Arrival Position (VDFAP) distribution, which can be viewed as a generalization of symmetric alpha-stable (S$\alpha$S) distribution with stability parameter $\alpha=1$. In addition, VDFAP distribution has a clear physical interpretation when we consider first-hitting problems of particles following Brownian motion with a driving drift. Inspired by the Fourier relation between the probability density function and CF of Student's $t$-distribution, we extract an integral representation for the VDFAP probability density function. Then, we exploit the Hankel transform to derive a closed-form expression for the CF of VDFAP. From the CF, we discover that VDFAP possesses some interesting stability properties, which are in a weaker form than S$\alpha$S. This calls for a generalization of the theory on alpha-stable distributions.

翻译：近期研究表明，稳定分布在建模重尾或脉冲噪声方面具有显著优势。若对应特征函数存在闭式表达式，则可极大促进概率分布稳定性的研究。本文探索了一类新型分布——垂直漂移首次到达位置分布，该分布可视为稳定参数α=1的对称α稳定分布的推广。进一步地，当考虑具有驱动漂移的布朗运动粒子的首次击中问题时，VDFAP分布具有清晰的物理意义。受学生t分布概率密度函数与特征函数之间傅里叶关系的启发，我们提取了VDFAP概率密度函数的积分表达式，进而利用汉克尔变换推导出其特征函数的闭式解。通过特征函数发现，VDFAP具有若干有趣的弱稳定性特性，其稳定程度弱于对称α稳定分布。这一发现呼吁对α稳定分布理论进行推广。