The path length of random skip lists - Institut für Analysis und ...

With a = Em>1 Q ,"- 1 .
In order to get the asymptotic behaviour of [z'- ' ]H(z) for n oo we extend
the contour of integration to a large rectangle and shift the left side to the line
Re z = 1 + E, E > 0. Then [z" -1 ]H(z) equals the sum of residues at the additional
poles included in the contour with an error term of order O(n 1+ E) (compare (3] for
technical details of the estimate of the error term) . In our instance the additional
poles originate from the second integral, and they are located at z = 2 (triple) resp .
z = 2 + Xk, k E Z \ {0} (simple) . The computation of the residues can be performed
by hand, or, more preferably, e .g . by Maple, and we find for the residue at z = 2
2(Q - 1)2
+ Hn2) 1
n + 1 ~H,2,_i Hn
1 (1 + 2
2 L 2 L2 L )
+ 3 + L2 - 2(a +,3) + 2L + (Q 1 I )L }'
(2 .31)
where L = log Q, H,,, resp . H(,,,2) are the harmonic numbers of first resp . second order,
and ,0= E 1 2 .
m>1
(Qm- 1 )
The asymptotic equivalent of (2 .31) is
2 g 2
2(Q - 1) 2 n12 Q n + n 2 logQ
n C L 2 L l
2 1 1 'Y 'Y 'Y2
~2
1 1 l
+n (6+L2-a-~-2L L2+ 2L2+ 12L2+ 4L + 2(Q - IL) )
+O (n log e n)
. (2.32)
I
The first order poles at z = 2 + Xk, k E Z \ {0}, give a contribution of the form
n 2 68 (log Q n) + 0(n), (2 .33)
where 58(x) is a continuous periodic function of period 1 and mean zero . The Fourier
coefficients of 68(x) can be obtained explicitly from the residues at the above . mentioned
poles . They can be used to show that the amplitude of 68(x) is very small
(because of the exponential decrease of the F-function along . vertical lines) . Since
from the practical .point of view 6 8 (x) has almost no influence on the variance, we
omit these calculations here .
Combining (2 .32), (2 .33) and Proposition 1 we get the asymptotic equivalent of
Var(Xn,) according to (2 .11) . The reader should take notice of the fact that the term of
order n2 in E(X n )2 contains the square 6 2 (logQ n) of the fluctuating term 6 1 (logQ n),
where 61(x) has mean zero . Therefore we have
2
Var(Xn,) = (Q - - 2(a +)(3) +
[
1)2n2 2L + 12 2L + 6L2 (Q _ 1 1)L 2L 6 1
0
+n 2 64(logQ n) + 0(n 1 +6 ), E > 0, (2 .34)
where [ 62 ] o is the mean of 62 (x), and 64 (x) is a small periodic function of period 1
and mean zero .