On the Internal Path Length of d–dimensional Quad Trees

the depth of the root is defined to be one; then Y 1 = 1, Y 2 = 3, . . . Since the
subtrees of a random quad tree are again random quad trees the following
recursion for the internal path length holds in distribution
Y n D =
2 d −1 ∑
k=0
Y (k) + n (12)
I (n)
k
where (Y (k)
i ) are independent copies of Y i and {(Y (k)
i ), k = 0, . . . , 2 d −1}, I (n)
are independent. We define Y 0 := 0. The expectation of the internal path
length Y n is given in (3). The normalized version X n of Y n given by
satisfies the modified recursion
X n := Y n − IEY n
n
(13)
X n D =
2 d −1 ∑
k=0
I (n)
k
n X(k) I (n)
k
+ C n (I (n) ) (14)
where (X (k)
i ) are independent copies of X i , further {(X (k)
i ), k = 0, . . . , 2 d −
1}, I (n) are independent and
⎛
⎞
C n (i) := 1 + 1 2∑
d −1
⎝ IEY ik − IEY n
⎠ (15)
n
k=0
for i = (i 0 , . . . , i 2 d −1) with ∑ i k = n − 1.
3 Limit theorem for the internal path length
In order to obtain a limiting form of the recursion (14) we introduce the
simplex
and the entropy functional
T 2 d −1 := {x ∈ [0, 1] 2d |
2 d −1 ∑
i=0
C : T 2 d −1 → IR, C(x) := 1 + 2 d
6
x i = 1} (16)
2 d −1 ∑
i=0
x i ln x i (17)