We have worked in a lecture (about the optional stopping theorem) with the following two inequalites:
$\mathbb{E}[T \mid X_0] \leq \frac{X_0}{c}$
and
$\mathbb{E}[T ] \leq \frac{\mathbb{E}[X_0]}{c}$,
where $T$ is the stopping time $T:=\min\{t \geq 0 \mid X_t = 0\}$ for a discrete-time random process $(X_t)_{t \geq 0}$ that takes non-negative integer values.
I am not sure whether these two inequalities are equivalent, one implies the other or whether they are completely unrelated? It would be nice if you could give me an intuition and also a formal proof or at least a hint how such a proof could work. Also, I am interested in whether (if equivalence holds) this is due to the definition of the random variables or whether it does hold for (almost) all random variables?
Thank you very much!