论文标题
离散随机步行和Vandermonde矩阵的分布
Distribution of Shifted Discrete Random Walk and Vandermonde matrices
论文作者
论文摘要
在这项工作中，我们设置了最终时间生存概率$φ（u+1）$的生成函数，其中$$φ（u）= \ mathbb {p} \ left（\ sup_ {n \ geqslant 1}}} $ u \ in \ Mathbb {n} _0，\，\，κ\ in \ mathbb {n} $和随机步行$ \ left \ left \ left \ {\ sum_ {\ sum_ {i = 1}^{n} x_i，\，\ in \ in \ mathbb {n} $ right priancation $ right and sisters $ sisters $价值非负和整数。我们还通过某些多项式的根部给出$φ（u）$的表达式。基于经过验证的理论语句，当随机变量$ x_i $允许bernoulli，几何和其他一些分布时，我们给出了几个有关$φ（u）$及其生成函数表达式的示例。
In this work we set up the generating function of the ultimate time survival probability $φ(u+1)$, where $$φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right)$$ and $u\in\mathbb{N}_0,\,κ\in\mathbb{N}$, and the random walk $\left\{\sum_{i=1}^{n}X_i,\,n\in\mathbb{N}\right\}$ consists of independent and identically distributed random variables $X_i$, which are non-negative and integer valued. We also give expressions of $φ(u)$ via the roots of certain polynomials. Based on the proven theoretical statements, we give several examples on $φ(u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, Geometric and some other distributions.
