I have found a lot of the confusion people have about the Central
Limit Theorem comes from lack of exactnes.
CLT = For large random samples the sampling distribution of the mean
is approx. normal
In VP we take it that a random sample applies. The mean or average
profit is what converges to the normal - nothing related to
individual hands apply except for the "randomn" part.
How large is large? This depends on the underlying distribution of
each individual sample or in VP each hand. The more the skewness in
the individual distribution in each hand the longer it will take the
distribution of the mean to be "normal". The skewness of the sample
mean payout for an individual game is dependend on the sample size or
number of hands and when this gets close to zero a large sample is
reached. Other measures of nonnormalness could be used.
But I think may players are really interested in their aggregate play
and convergence of realized payouts to the expected payout for
individual games. This is the Law of Large Numbers. The trouble is
that while the mean converges to the EV for any game the sum or
aggregate of all payouts does not converge.
This last part in my view is an interesting paradox in the practical
application of vp