“Lottery Voting: A Thought Experiment” is a 1995 article by Akhil Reed Amar, perhaps the most influential Constitutional Law theorist in the US.
In the article Amar proposes using the lottery to determine representation only after a standard election campaign has determined the percentage of support each candidate has received.
In this case then, the ‘statistical representation’ (which he champions) would be only of those citizens willing to go through the process of standing for election.
This is much less representative — and less attractive to me — than eliminating campaigning altogether… and using only two parameters to enter the lot: 1.) registration for it, indicating a willingness to serve if chosen; 2.) a simple civics test (such as the Naturalization Test in the U.S.).