Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Abstract

In this paper, we set up the distribution function $$ \varphi(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right), $$ and the generating function of $\varphi(u+1)$, 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 $1-\varphi(u)$ or survival probability $\varphi(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\varphi(u)$ 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.

Figures (2)

• Figure 1: Lines $1+n$, $1+2n$, $1+3n$, and random walk $\sum_{i=1}^{n}X_i\mathbbm{1}_{\{i\mod 2=1\}}+\sum_{i=1}^{n}Y_i\mathbbm{1}_{\{i\mod 2=0\}}$, where $\mathbb{P}(X_i=0)=0.3$, $\mathbb{P}(X_i=1)=0.1$, $\mathbb{P}(X_i=5)=0.6$ and $\mathbb{P}(Y_i=0)=0.8$, $\mathbb{P}(Y_i=1)=0.1$, $\mathbb{P}(Y_i=10)=0.1$, and $n$ varies from $1$ to $20$.
• Figure 2: Roots of $s^{50}=G_{S_{10}}(s)$, when $X_k\sim\mathcal{P}(k/(k+1)+4,\,0)$, $k\in\{1,\, 2,\,\ldots,\, 10\}$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3