Consider the random variable:
the number of occurences of three consecutive 2's surounded by 1s (i.e. the \
factor 12221 )
in a (uniformly) drawn n-letter word in the alphabet {1,2}
n
The mean is:, - 1/8 + ----
32
13 27 n
The variance is:, - --- + ----
128 1024
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 (r - 1) (1452 r - 1937)
(2 r)! |1 + ------------------------- +
| 729 n
\
4 3 2 \
r (r - 1) (3513840 r - 35152920 r + 81854525 r + 8684591 r - 69483282)|
-------------------------------------------------------------------------|
2 |
5314410 n /
/ r
/ (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:
/ 3 2
11 1/2 1/2 | (r - 1) (5324 r - 37279 r + 67839 r - 10362)
--- (2 r)! 1024 3 |r - 1 + ----------------------------------------------
432 \ 8019 n
6 5 4 3
+ (r - 1) (7730448 r - 157239016 r + 1049602455 r - 2752418194 r
\
2 / 2 | / r
+ 1947122937 r + 1982085962 r - 1523789160) / (58458510 n )| / (r! 2
/ / /
1/2
n )
In particular it is (as expected) asymptotically normal, but we have an even\
finer asympotics for the alpha coefficients.
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