# Probability of ruin in the Sparre Andersen renewal risk model

In 1957, Erik Sparre Andersen proposed using a renewal risk model to describe the behavior of the insurers surplus

$$U(t)=u+ct-\sum\limits_{i=1}^{\Theta(t)}Z_i, \quad t \geqslant 0$$

where:

• $u \geqslant 0$ denotes the initial insurer's surplus;

• $c > 0$ denotes the premium rate per unit of time;

• the cost of claims $Z_1, Z_2,\ldots$ are independent copies of a nonnegative random variable $Z$;

• the inter-occurrence times of claims $\theta_1, \theta_2, \ldots$ are another sequence of independent copies of a nonnegative random variable $\theta$ which is not degenerate at zero;

• the sequences $\{Z_1, Z_2, \ldots\}$ and $\{\theta_1, \theta_2, \ldots\}$ are mutually independent;

• $\Theta(t) = \#\{n \geqslant 1: T_n \in[0,t]\}$ is the renewal process generated by random variable $\theta$, where $T_n = \theta_1 + \theta_2 + \cdots + \theta_n$.

It is known that, for $\mathbb{E}Z-c\mathbb{E}\theta<0$ and $u=0$, the ruin probability

$$\psi(u)=\mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(Z_i-c\theta_i)>u\right)$$ has the following expression $$\psi(0)=1-\exp\left\{-\sum_{n=1}^{\infty}\frac{1}{n}\mathbb{P}\left(\sum_{i=1}^{n}(Z_i-c\theta_i)>0\right) \right\}.$$

It is also known that $\psi(0)=1$ for $\mathbb{E}Z -c\mathbb{E}\theta\geqslant0$.

I want to see how the series

$$\sum_{n=1}^{\infty}\frac{1}{n}\mathbb{P}\left(\sum_{i=1}^{n}(Z_i-c\theta_i)>0\right)$$

diverges as $\mathbb{E}Z-c\mathbb{E}\theta$ approaches zero.

Let $X$ denote a r. v. such that $\mathbb{E}X=-\varepsilon$, where $\varepsilon>0$ and $\lim_{\varepsilon\to0}\mathbb{E}X^2>0$.

Can you show that

1. $\mathbb{P}(X>0)>0$?

Or in general

1. $\mathbb{P}\left(\sum_{i=1}^{n}X_i>0\right)>0$ for all $n\in\mathbb{N}$, where $X_i$ are independent copies of $X$.

It is natural to expect that both statements may hold only as $\varepsilon$ approximates zero.

$\newcommand{\vp}{\varepsilon}$ $\newcommand{\P}{\mathbb P}$ $\newcommand{\E}{\mathbb E}$
It appears that your $X$ denotes a random variable $X_\vp$ depending on $\vp$. Suppose then that $\P(X_\vp=-1/\vp)=\vp^2=1-\P(X_\vp=0)$ for $\vp\in(0,1)$. Then $\E X_\vp^2=1\to1>0$ as $\vp\downarrow0$, but $\P(X_\vp>0)=0$ for $\vp\in(0,1)$. So, your conjectures 1) and 2) are in general false.
• Ok. As we see, there are distributions for which conjectures fails, however for some it works. Let's assume that the limit distribution is not degenerate as $\varepsilon \to 0$, i.e. $\lim_{\varepsilon \to 0}P(X_{\varepsilon}=x_{\varepsilon , i})<1$ for all $x_{\varepsilon , i}$. – Fancier of Mathematica Feb 1 '18 at 6:58