I would like to place a lower limit under two consecutive \sums, somewhere in between. So I have \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} and would like to place j>k in the lower limit between the two sums. I believe the multiple integral operator has this kind of functionality.
[Tex/LaTex] Limits under multiple \sum
math-modemath-operators
Related Solutions
Maybe this? with a little help from \mathclap and stackengine. EDIT to replace \ldots with \dots and \cdots, per Mico's request. I hope I interpreted his request in the right way...
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{stackengine}
\stackMath
\begin{document}
$\displaystyle x \wedge y =\qquad
\sum_{\mathclap{1 \leq i_1 < \dots < i_p \leq n}}\quad
\sum_{\stackunder[5pt]{}{\scriptstyle\mathclap{1 \leq j_1 < \dots < j_q \leq n}}}
a_{i_1 \dots i_p} b_{j_1 \dots j_q}
\; \underbrace{e_{i_1} \wedge \cdots \wedge e_{i_p} \wedge e_{j_1} \wedge \cdots \wedge e_{j_q}}_{\text{is sometimes 0}}$
\end{document}
Another solution, also provided by the mathtools package, is to use its \smashoperator macro. In fact, by writing \smashoperator[r]{...}, one can arrange that the material overlaps only on the right, rather than on both sides. (Observe that the word "Note" does not overlap with the material below the summation sign.)
Note that I would also create a macro, called (say) \VaR, to typeset the string VaR with upright letters. And, do note that there's no point in encasing P(X)\cdot x in curly braces.
\documentclass{article}
\usepackage{mathtools}
\newcommand\VaR{\mathrm{VaR}}
\begin{document}
\begin{equation}
\text{Note:}\smashoperator[r]{\sum_{X\leq\VaR_\alpha(X)}} P(X)\cdot x
\end{equation}
\end{document}
Addendum to address the OP's follow-up question: You may achieve your typesetting objective by (a) switching to the basic, i.e., centered version of the \smashoperator directive and inserting a suitable "kern" at the start of the subscript material. I know it's not fruitful to argue about tastes, but in my opinion the result of right-shifting the subscript material is anything but good-looking.
\documentclass{article}
\usepackage{mathtools}
\newcommand\VaR{\mathrm{VaR}}
\begin{document}
\[
\text{Note:}\smashoperator{\sum_{\mkern70mu X\leq\VaR_\alpha(X)}} P(X)\cdot x
\]
\end{document}
Best Answer
Something like:
\mathop{\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}}_{j>k}maybe?
Looks like this:
Or if you want it on the same line as j=1 and k=1, then maybe:
\mathop{\sum^{\infty}\sum^{\infty}}_{j=1\ j>k\ k=1}