Thanks for opening this question which has bothered me for a while.
Given random variables $e_{it}$ and $a_i$ for $i=1,...,N$ and $t=1,...,T$ with the following properties: \begin{align} \frac{1}{T}\sum_{t=1}^T e_{it} &= o_p(1) \ \ \forall i\in{1,...,N}\\ \frac{1}{N}\sum_{i=1}^N |a_i| &= O_p(1) \end{align} WTS \begin{align} \frac{1}{NT}\sum_{i=1}^N\sum_{t=1}^Te_{it}a_i \stackrel{p}{\to} 0\ \ as \ \ N,T\to\infty \ \ jointly \end{align}
Attempt: \begin{align} \frac{1}{NT}\sum_{i=1}^N\sum_{t=1}^Te_{it}a_i & \leq \frac{1}{NT}\sum_{i=1}^N \sup_{i\in {1,...,N}} \left\{\left|\sum_{t=1}^Te_{it} \right|\right\} \left|a_i\right| \\ & = \sup_{i\in {1,...,N}}\left\{\left|\frac{1}{T}\sum_{t=1}^Te_{it} \right|\right\} \frac{1}{N}\sum_{i=1}^N \left|a_i\right| \end{align}
It seems like I need uniform convergence like \begin{align} \sup_{i\in\{1,...,N\}} \left| \frac{1}{T} \sum_{t=1}^Te_{it} \right| \stackrel{p}{\to} 0 \ \ as \ \ N,T \to \infty \ \ jointly \end{align} However, I can't prove the uniform convergence since $i\in\{1,2,...\}$ is not compact. (The theorem is here: Uniform consistency of a sum of functions with random weights?)
Is there anyway to show it? Some additional information may be useful but I am not sure how to use: ($a_i$ are i.i.d across $i$, $e_{it}$ and $e_{jt}$ are not correlated for $i\ne j$).
(A related question: Infinite sum of little o-p terms)
I really appreciate it if you have any thoughts! Thanks a lot!