I have the following quantity that I would like to control:

> Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and bounded variance, for
 all $N\geq N_0$ large  and $t\in (1/N, t_0)$ where $t_0$ is sufficiently
 small, I would like to get the tail estimate  $$\mathbb{P} \left\{\inf_{1\leq k \leq tN} \sup_{tN
 \leq l \leq N} \left( \frac{1}{l-k}\sum_{i=k+1}^l Z_i \right)\leq t
 \right\} \leq C t^\alpha$$
for some positive $C, \alpha$. The same estimate can be rewritten as 
$$\mathbb{P} \left\{\inf_{1\leq k \leq tN} \sup_{tN
 \leq l \leq N} \left(\sum_{i=k+1}^l Z_i \right) - t(l-k) \leq 0
 \right\} \leq C t^\alpha.\quad (\star)$$

To simplify the question, we can look at
$$\mathbb{P} \left\{\sup_{1 \leq l \leq N} \left( \frac{1}{l}\sum_{i=1}^l Z_i \right)\leq t \right\} \leq C t^\alpha.$$
(I know the distributions of $Z_i$'s, if this is helpful). 

Is there a name for this type of inequality where we look at the maximum of the averages (or the sum of i.i.d. random variables but we can not move the constant to the other side, like in $\star$ above).

I found a related general results in [this][1] paper by Chung; here the mean zero random variables are only assumed to be independent. With his notation, $S_n^* = \max_{1\leq k\leq n} |S_n|$, and $s_n = \text{Var}[S_n]$ which is  $Cn$ in the i.i.d. case, we have

**Theorem 2.** If $g_n \downarrow 0$ and 
$$g_n^{-1} = O((\log_2 s_n)^{1/2})$$
then we have 
$$\mathbb{P}(S_n^* < g_ns_n) = (1+o(1)) \exp\left(-\frac{\pi^2}{8g_n^2}.\right)$$

Is there a simpler inequality of this type for i.i.d. random variables? The proof of this inequality in his general setting is quite technical.


**Background:**
The original event that I was trying to estimate is 
$$\left\{\inf_{1\leq k \leq tN} \sup_{tN
 \leq l \leq N}\sum_{i=k+1}^l X_i - Y_i  \leq 0\right\}$$
where $X_i \sim \exp(\rho)$, and $Y_i \sim \exp(\rho- t)$ all independent of each other. 

Like Kolmogorov or Doob's maximal inequality, maybe it is helpful to center the random variables; by defining
$Z_i = X_i - Y_i - \mathbb{E}[X_i - Y_i] $, we get the centered version 
$$\left\{\inf_{1\leq k \leq tN} \sup_{tN
 \leq l \leq N}\sum_{i=k+1}^l \left(Z_i - \frac{t}{\rho(\rho-t)} \right)   \leq 0 \right\},$$
and this boils down to estimate the event in my question above.

**Final remark:**
One way to get some kind of tail estimate is to go to Brownian motion using Donsker's theorem, and we could obtain 
$$\limsup_{N\rightarrow \infty} \mathbb{P} \left\{\inf_{1\leq k \leq tN} \sup_{tN
 \leq l \leq N} \left( \frac{1}{l-k}\sum_{i=k+1}^l Z_i \right)\leq t
 \right\} \leq C t^\alpha$$ for all $t\in (0, t_0)$. In this case, the $N_0$ would be dependent on $t$ so instead of $``N\geq N_0"$ we have to use $``\limsup_N"$, and I am trying to avoid this.


  [1]: https://www.ams.org/journals/tran/1948-064-02/S0002-9947-1948-0026274-0/S0002-9947-1948-0026274-0.pdf
  [2]: https://i.sstatic.net/SQm3x.png