4
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

$\newcommand{\vp}{\varepsilon}$ $\newcommand{\P}{\mathbb P}$ $\newcommand{\E}{\mathbb E}$

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.

$\endgroup$
  • $\begingroup$ 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 $x_{\varepsilon , i}$. $\endgroup$ – Fancier of Mathematica Feb 1 '18 at 6:58

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.