Let $X_0,X_1,...$ be a sequence of i.i.d $\mathbb{R}$-valued random variables with $$\mathbb{E}[X_i] = \alpha.$$ Define $Z_0 =0$ and for $n\geq 0$, $$Z_{n+1} = \max\{0, Z_{n} + X_n \}.$$ In words, $Z_n$ is a random walk with a reflection at $0$. Define the first hitting time $$T = \inf\{n \geq 0 \mid Z_{n} \geq \delta \}.$$ I want to bound $\mathbb{E}[T]$ under various conditions. What tools can I use to do this?

For instance I want to prove a lower bound on $\mathbb{E}[T]$ when $\alpha < 0$ and I want to make an upper bound on $\mathbb{E}[T]$ when $\alpha > 0$. I have no idea where to start on this, what theorems to use, etc...


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