# Do adjacent order statistics from a uniform distribution converge almost surely?

In class we found the asymptotic distribution of the sample median of iid uniform random variables. We did this by cases, considering odd-numbered and even-numbered samples separately. It occurred to me that one case would suffice if we could establish that adjacent order stats converge almost surely, and my intuition says that they do. (I'm guessing this is true whenever dealing with iid variables bounded in probability, but I'm specifically interested in uniform r.v.'s.)

I tried my hand at proving this, but so far I've come up short. I tried arguing as follows: Consider $X_i\stackrel{iid}\sim \text{Unif}(0,1)$, let $\epsilon>0$, then for any order stat $X_{(m)}$, a new draw has probability $\epsilon$ of falling in the region $(X_{(m)},X_{(m)}+\epsilon)$; and since $\sum_{k=0}^\infty (1-\epsilon)^k < \infty$, we have the desired convergence. But then I realized this fails since $X_{(m)}$ can move around, and so even if a new draw falls in the region $(X_{(m)},X_{(m)}+\epsilon)$, subsequent draws might pull $X_{(m)}$ and $X_{(m+1)}$ farther apart again.

This is a bad way to set it up because the (m)th order statistic of a uniform R.V. converges to zero. If you take, for example, the 30th order statistic, then with nine quadrillion samples, it's gonna be close to zero. The median is a quantile, and its estimate is a different order statistic depending on the sample size. So, I'd advise you to start with the floor(n/2) and floor(n/2) + 1 order statistic. In general, though, this tactic seems a little cumbersome because those two quantities are not independent, and their difference is hard to get a PDF or a bound for. (Maybe it's not hard for everyone here on CV, but it seems harder than what you probably did in class.)
• Or, if $n$ is large, one could use the concept of fractional order statistic to get an approximation. See amazon.com/Order-Statistics-Herbert-David/dp/0471389269 Oct 10, 2017 at 21:37
• Yes, this is only tangentially related to what we did in class. I take your point about defining this in terms of a quantile. I agree it'd be very difficult to get a joint pdf for these quantities, but I am wondering whether I really need to do that. For example, I was thinking along the following lines: let $Y_n$ be the maximum "gap" size after $n$ draws; that is, $Y_n=\max_{i=1,\cdots,n-1} (X_{(i+1)}-X_{(i)})$. Maybe I could show $Y_n\xrightarrow{a.s.}0$? And that would suffice?