On the Internal Path Length of d–dimensional Quad Trees

By calculation as in the proof of Lemma 3.2
⎛
⎞
IEC 2 (〈U〉) = −1 + 4 2
d IE ∑
d −1
⎝ 〈U〉 2 i ln〈U〉 i
⎠
= −1 + 4 2∑
d −1
d 2
i,j=0
i=0
IE[〈U〉 i 〈U〉 j ln〈U〉 i ln〈U〉 j ].
The distribution of the factors 〈U〉 i 〈U〉 j ln〈U〉 i ln〈U〉 j only depends on the
number of digits in which the dual representations of i and j differ (see (5),
(6)). Therefore
⎛
⎞
2∑
d −1
IE ⎝ 〈U〉 i ln〈U〉 i
⎠
i=0
2
2
( )
d∑ d
= 2 d l h . (45)
h=0
h
l h can be calculated by first applying the functional equation of the logarithm.
This yields d 2 terms of the form
d−h ∏ d∏
u
∫[0,1] 2 d i
i=1 i=d−h+1
(u i (1 − u i )) ln ũ k ln ũ l dλ d (u)
with ũ k = u k for k ≤ d − h and ũ k = 1 − u k for k > d − h. Then distinguish
the cases 1 ≤ k, l ≤ d − h and d − h + 1 ≤ k, l ≤ d for k = l and k ≠ l
and finally 1 ≤ k ≤ d − h < l ≤ d. The arising integrals can be calculated
elementary. This implies the representation
( ( 2 d
) −1 [
v d = 1 − −1 +
3) 4 ( ) 2 d d∑
( ) (1 d h
]
s
d 2 h
3
h=0
h 2)
(46)
where
s h =
( d
3 − h 2
) 2
+ d ( ) 5
9 + 4 − π2
h.
6
Now a simplification with the help of Maple leads to the stated variance.
l 2 -convergence implies convergence of second order moments. We obtain as
Corollary the first order asymptotics of the variance of the internal path
length Y n .
14