Consider the random variable:
the number of occurrences of three consecutive 2's
in a (uniformly) drawn n-letter word in the alphabet {1,2}
The mean is:, - 1/4 + n/8
15 n
The variance is:, - 5/8 + ----
64
The asympotics to order 2, of the even alpha coefficients (the (2r)-th momen\
t about the mean divided by the r-th power of the variance)
as an expression in n and r is:
/ r (8464 r - 2843) (r - 1)
(2 r)! |1 + ------------------------- + r (r - 1)
\ 3375 n
4 3 2 /
(71639296 r - 287657504 r + 107356331 r + 552524681 r - 129963390) / (
/
2 \ / r
68343750 n )| / (r! 2 )
/ /
The asympotics to order 2, of the odd alpha coefficients (the (2r+1)-th mome\
nt about the mean divided by the (r+1/2)-th power of the variance)
as an expression in n and r is:
/
23 1/2 1/2 1/2 |
--- (2 r)! 5 64 3 |r - 1
450 \
3 2
(r - 1) (194672 r - 780183 r + 788722 r + 107412)
+ --------------------------------------------------- + (r - 1) (
232875 n
6 5 4 3
1647703808 r - 17068451616 r + 55772899985 r - 42096964320 r
\
2 / 2 | /
- 82388439343 r + 105281965926 r - 419701680) / (7859531250 n )| / (
/ / /
r 1/2
r! 2 n )
In particular it is (as expected) asymptotically normal, but we have an even\
finer asympotics for the alpha coefficients.
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