Silly statistics question - sci.math

Message from discussion Silly statistics question
Path: g2news1.google.com!postnews.google.com!p6g2000pre.googlegroups.com!not-for-mail
From: Daniel McLaury <daniel.mcla...@gmail.com>
Newsgroups: sci.math
Subject: Silly statistics question
Date: Tue, 31 Mar 2009 10:16:34 -0700 (PDT)
Organization: http://groups.google.com
Lines: 18
Message-ID: <d2d351f7-1064-4556-afea-40ec93968be7@p6g2000pre.googlegroups.com>
NNTP-Posting-Host: 69.231.60.78
Mime-Version: 1.0
Content-Type: text/plain; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
X-Trace: posting.google.com 1238519794 23807 127.0.0.1 (31 Mar 2009 17:16:34 GMT)
X-Complaints-To: groups-abuse@google.com
NNTP-Posting-Date: Tue, 31 Mar 2009 17:16:34 +0000 (UTC)
Complaints-To: groups-abuse@google.com
Injection-Info: p6g2000pre.googlegroups.com; posting-host=69.231.60.78; 
	posting-account=IypdrgkAAACU7jod879UbhndSrdgIgGP
User-Agent: G2/1.0
X-HTTP-UserAgent: Mozilla/5.0 (Windows; U; Windows NT 6.0; en-US; rv:1.9.0.8) 
	Gecko/2009032609 Firefox/3.0.8,gzip(gfe),gzip(gfe)

I've been thinking this over in the back of my mind for some time now
and finally I thought I'd just ask someone because I don't really have
the background to figure it out for myself.

Let X_1, X_2, ... X_N be i.i.d. random variables, and let Y_N = [ X_1
+ X_2 + ... + X_N ] / N.  Put f(N) = Median(Y_N).

If Var(X_i) is finite, then as N approaches infinity f(N) approaches E
(X_i).  I can prove that much.  I do not even know what happens when E
(X_i) exists and Var(X_i) does not.

What interests me is that, if this holds even for infinite variance,
then this is potentially a generalization of the expectation, since it
could presumably be defined in cases where the expectation is not.  Of
course, more likely it's either exactly equivalent to expectation or
it's not even defined even for some distributions with well-defined
expectations.  Anyway, I just wondered if anyone know how to work with
problems like this.