A potential way in which the Central
Limit Theorem (CLT) can break down via combinations of lots of independent
gamma distributed random variables
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The Gamma distribution has the the ‘summation’ property that
if for
and
the are
independent then .
So, suppose .
Then . Thus we appear to be
combining more and more (independent) random variables each of which has
smaller and smaller mean, ,
and variance, ,
so we might expect the Central Limit Theorem (CLT) to apply.
In fact, for the CLT to apply we need somewhat more onerous
regularity conditions to be satisfied, including a focus on and
(usually) that the distributions of the do
not change as changes (as well as being
of finite variance). The above example does not satisfy these amplified
regularity conditions because as changes the distribution
of each changes.
Although the means (and variances) get smaller and smaller (which you would
have thought would help with satisfying the CLT), each individual becomes
more and more skewed and has a greater and greater (excess) kurtosis, see e.g.
the Nematrian webpage on the Gamma distribution.