I have $m$ continues integer points on a line, randomly uniform select $n$ points from the $m$ point without replacement. Order the points ascendingly. Let the random variable $A_i$ is the position (coordination on the line) of the $i$th point. So, $P(A_i=k)=\frac{{k-1\choose i-1} {m-k \choose n-i}}{{m \choose n}} $
How to derive the tail inequality for this probability. The tail probability look something like this:
$P(|A_i - E(A_i)| > t) < \sigma$
I want the bound ($\sigma$) to be as tight as possible. The Chebyshev inequality is too loose.
Updated: Some supplement about the question: http://www.math.uah.edu/stat/urn/OrderStatistics.pdf