From the amsmath documentation (texdoc --view amsldoc.pdf):
If the new operator should have subscripts and superscripts placed in “limits”
position above and below as with lim, sup, or max, use the * form of the \DeclareMathOperator command:
\DeclareMathOperator*{\Lim}{Lim}
Also, the \mathop is not needed. After all, you declare a math operator with \DeclareMathOperator[*].
I refer to „What is the difference of \mathop, \operatorname and \DeclareMathOperator?“ for further reading on this topic.
\DeclareMathOperator*{\<new op>}{<stuff that should be typeset as an op with limits>}
This answers the question why the limits aren’t where you have expected them to be.
But \bicup is already an operator you should do
\newcommand*{\Union}{\bigcup}
instead to have a more descriptive name for it (which is a good idea nonetheless).
The \DeclareMathOperator* pair of macros is to declare new operators like the already defined \lim, \sum, and so on.
Code
\documentclass{aomart}
\newcommand*{\Union}{\bigcup}
\begin{document}
\[ \Union_{i \in I} \bigcup_{i \in I} \]
\end{document}
Output

\[ \]),\[\bigcup_{\i \in I}\]should result in subscript underneath the symbol as expected. Can you give a full MWE? – Corentin Mar 04 '13 at 18:53\DeclareMathOperator*and\DeclareMathOperator? – Popopo Mar 04 '13 at 19:09\bigcupit works perfectly; if you want a new name for it, just do\newcommand{\Union}{\bigcup}– egreg Mar 04 '13 at 20:19