A handy little macro from comp.text.tex (2000) buried in one of the massive “Giant List O’ Symbols” documents on TUG. This makes a \perp with two vertical lines instead of one.
\newcommand\independent{\protect\mathpalette{\protect\independenT}{\perp}}
\def\independenT#1#2{\mathrel{\rlap{$#1#2$}\mkern2mu{#1#2}}}