The uniform asymptotic negligibility (UAN) assumption is well know in probability theory. In my case, I have a definition of (UAN) for moving-average (MA) processes.
Let $(X_n)_{n\geq 1}$ a sequence of MA processes: $$X_{t;n}= \sum_{j =1}^n \psi_{j;n} \varepsilon_{t-j;n}, \quad (t \in \mathbb{Z}, n \in \mathbb{N})$$ We say that $(X_n)_{n\geq 1}$ satisfyes UAN condtion if \begin{equation}\tag{UAN} \underset{n \to \infty}{\lim} \underset{1\leq j \leq n}{\max} |\psi_{j;n}|=0 \end{equation} It turns out that at a given moment, the coefficients $\psi_{j;n}$'s are random variables. So I would like to know how to adapt the UAN definition. My first attempt is: \begin{equation}\tag{UAN 1} \underset{n \to \infty}{\lim} \underset{1\leq j \leq n}{\max} P( |\psi_{j;n}| > \epsilon )=0\quad (\forall \, \epsilon > 0) \end{equation}
Another alternative is \begin{equation}\tag{UAN 2} \underset{n \to \infty}{\lim} P\big( \underset{1\leq j \leq n} {\max} |\psi_{j;n}| > \epsilon \big)=0\quad (\forall \, \epsilon > 0) \end{equation} Or maybe: \begin{equation}\tag{UAN 3} P\big(\underset{n \to \infty}{\lim} \underset{1\leq j \leq n} {\max} |\psi_{j;n}| = 0 \big)=1 \end{equation}
Honestly, I'm quite confused on which definition would be the most appropriate.
Some reference, help!