\documentclass{article}

\usepackage{pstricks} % To use the standard "color" package with PSTricks
\usepackage{multido}
\input{random.tex}         % From Donald Arseneau (on macros/generic on CTAN)

\newdimen\dX
\newdimen\dY
\newdimen\Coor

% Random walk in the unit square (#1 = number of steps)
\def\RandomWalk#1{%
\setrandim\dX{0pt}{1pt}
\setrandim\dY{0pt}{1pt}
\psdot[dotstyle=*,dotsize=6pt](\pointless\dX,\pointless\dY) % Initial point
\pscustom{%
  \moveto(\pointless\dX,\pointless\dY)      % Move to initial point
  \multido{\i=1+1}{#1}{%
    \MoveCoordinate{\dX}
    \MoveCoordinate{\dY}
    \lineto(\pointless\dX,\pointless\dY)}}} % Line between old and new position

% Computation of new horizontal or vertical position of point
\def\MoveCoordinate#1{%
\setrandim\Coor{-0.06pt}{0.06pt}          % The move
\advance #1 by \Coor                      % New position
% To stay in the square...
\ifdim#1>1pt
  #1=1pt
\fi
\ifdim#1<0pt
  #1=0pt
\fi}

\pagestyle{empty}
\parindent=0pt

\begin{document}

\psset{unit=11,dimen=middle}
\begin{pspicture}(1,1)
  \psframe[linewidth=0.1pt](1,1)
  \RandomWalk{300}
  \psset{linecolor=red}
  \RandomWalk{300}
  \psset{linecolor=green}
  \RandomWalk{300}
\end{pspicture}

\end{document}
