论文标题
由不同的随机变量和毁灭理论中的应用产生的转移离散随机步行的分布
Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory
论文作者
论文摘要
在本文中，我们设置了分布函数$$φ（u）= \ Mathbb {p} \ left（\ sup_ {n \ geqslant 1} \ sum_ {i = 1}^{n}^{n} {n} \ left（x_i-i-κ\ right） $ u \ in \ MATHBB {n} _0 $，$κ\ in \ MATHBB {n} $，随机步行$ \ left \ left \ {\ sum_ {\ sum_ {i = 1}^{n} x_i，n} x_i，n \ n \ in \ mathbb {n} n}分布以及整数值和非阴性随机变量$ x_1，\，x_2，\，\，\ ldots $是独立的。这项研究概括了最近的两项作品，其中$ \ {κ= 1，\，n \ in \ mathbb {n} \} $和$ \ {κ\ in \ athbb {n}，\，\，n = 1 \} $。提供的总和的顺序$ \ left \ {生存概率$φ（u）$。验证获得的理论陈述我们演示了几个计算概率$φ（u）$及其生成功能时，当$ \ {κ= 2，\，\，n = 2 \} $，$ \ {κ= 3，\ 3，\，\，n = 2 \}我们还猜想了某些矩阵的非象征性。
In this paper, we set up the distribution function $$ φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right), $$ and the generating function of $φ(u+1)$, where $u\in\mathbb{N}_0$, $κ\in\mathbb{N}$, the random walk $\left\{\sum_{i=1}^{n}X_i, n\in\mathbb{N}\right\},$ consists of $N\in\mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,\,X_2,\,\ldots$ are independent. This research generalizes two recent works where $\{κ=1,\,N\in\mathbb{N}\}$ and $\{κ\in\mathbb{N},\,N=1\}$ were considered respectively. The provided sequence of sums $\left\{\sum_{i=1}^{n}\left(X_i-κ\right),\,n\in\mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-φ(u)$ or survival probability $φ(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $φ(u)$ and its generating function when $\{κ=2,\,N=2\}$, $\{κ=3,\,N=2\}$, $\{κ=5,\,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.
