At the end of my previous post I claimed that proving that tends to when tends to infinity and was *unpleasant*. It appears that I was wrong, and I would like to show here a simple proof that was suggested to me by my colleague Philippe Rigollet.

The key observation is that the term is the probability that an hypergeometric random variable takes value . Hypergeometric distributions are nasty, but when is of larger order than (which is the case here), they are well approximated by a simple binomial . In the present case this suggests the upper bound The latter inequality can be proved formally using the identity Now one can simply use the inequalities and to obtain that the original *unpleasant* sum is upper bounded by which clearly tends to when tends to infinity and .

## By Philippe Rigollet January 17, 2013 - 3:58 pm

I’ll try myself at posting my two (extra) cent about this proof with latex on WordPress.

The formula can be written as where has a hypergeometric distribution with the following parameters: is the population size, is the number of success and is the number of draws. I like to view this distribution as an experiment where one puts pebbles in slots at random (with only one pebble per slot) and then denotes the number of pebbles in the slots numbered through . This distribution enjoys a remarkable property, sometimes referred to as negative association: the fact that a pebble is in one of the first slots decreases the probability that another pebble is in one of these first slots (indeed one slot is already taken). Let’s see how we can use this property:

First, . We now show that each term tends to 1.

Let’s begin with the expected value:

where indicates if pebble is in one of the first slots. Now, negative association becomes useful. It implies that . But each is a Bernoulli random variable with parameter so that . For , we find that this quantity tends to 1 and .

We now deal with . A sufficient condition for this event to hold is that all the pebbles are in slots numbered at least . Therefore . By a union bound .