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

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

Or in general

- $\mathbb{P}\left(\sum_{i=1}^{n}X_i>0\right)>0$ 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.