Paley Zygmund inequality states that:
$$P(Z> \theta E[Z]) \ge (1-\theta)^2\frac{E[Z]^2}{E[Z^2]}.$$
Throughout the internet and text books, it is stated for a non-negative random variable Z (i.e. $Z\ge 0$). But, it seems the proof straightly follow if we only assume $E[Z]>0$. Indeed, the first step is
$Z= Z\mathbf{1}_{Z\le \theta E[Z]} + Z\mathbf{1}_{Z> \theta E[Z]}$ hence
$E[Z] \le \theta E[Z] + E[Z\mathbf{1}_{Z> \theta E[Z]}]$. So far, no assumptions over $Z$.
Next we apply C.S (which also doesn't require assumptions on $Z$): $E[Z] \le \theta E[Z] + \sqrt{E[Z^2]P(Z > \theta E[Z])}$. Which leads to: $$\sqrt{P(Z>\theta E[Z])} \ge (1-\theta)\frac{E[Z]}{\sqrt{E[Z^2]}}.$$
So far there were no assumptions on $Z$ (other than $E[Z^2]\ne 0$, which follows from $E[Z]>0$). Now we want to take the square of both sides, which indeed requires the two sides to be positive, which only requires $E[Z]>0$.
Why is Paley-Zygmund then, formulated to positive random variables? Does anyone know a reference that states it without the positivity requirement?