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finite divisibility in probability
Posted:
Apr 25, 2005 11:00 AM
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Feller's famous two volumes have a chapter on
infinite divisibility, and so do all other books.
A probability distribution F on the real line is infinitely
divisible precisely if for every positive integer n, there
exists a probability distribution G such that if X_1, ..., X_n
are independent random variables, the distribution of each of
which is G, then the distribution of their sum is F.
Everybody knows that lots of probability distributions are
infinitely divisible and lots of others are not.
What about finite divisibility? Given a probability distribution
F and a positive integer n, is there some probability distribution
G such that if X_1, ..., X_n are independent random variables, the
distribution of each of which is G, then the distribution of their
sum is F? Just omit the quantifier "for every n".
Is the uniform distribution on [0,1] "twice" divisible?
I.e., is there some distribution G such that the sum of
two independent G-distributed random variables is uniformly
distributed on [0,1]? Probably a trivial exercise, but
substitute other distributions for that one and plug in other
values of n than 2.
If a distribution is infinitely divisible then its sequence
{ k_n : n = 1, 2, 3, ... } of cumulants has the property that
for every number t between 0 and 1, the sequence
{ tk_n : n = 1, 2, 3, ... } is also the sequence of cumulants
of some probability distribution. Since the fourth central
moment is k_4 + 3 k_2^2 and must be non-negative, we must have
k_4 >or= -3k_2^2. In the (k_2, k_4)-plane, look at the straight
line from (k_2, k_4), the pair whose components are the second
and fourth cumulants of some probability distribution, to the
origin (0, 0). Somewhere along the way from (k_2, k_4) to
(0,0), you reach a region not satisfying the inequality
k_4 >or= -3k_2^2. Therefore, any distribution whose fourth
cumulant is negative fails to be infinitely divisible. But
suppose that for t = 1/2, i.e., for the pair (k_2, k_4)/2,
the inequality is still satisfied. Is the distribution "twice
divisible"? In some cases yes, obviously. In other cases,
other, similar inequalities fail. (Generally, if _any_ even-
numbered cumulant is negative, infinite divisibility will fail).
So is there a corpus of literature on finite divisibility?
Is it within the literature on infinite divisibility, being
considered merely a handmaiden of that topic? -- Mike Hardy
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