On the Internal Path Length of d–dimensional Quad Trees

= 1 + 2 d+1 (∫ 1
0
) d−1 ∫ 1
u du u ln u du
0
= 1 + 4(−1/4) = 0 (31)
so T is a well defined mapping T : M 0,2 → M 0,2 .
To prove contractivity let µ, ν ∈ M 0,2 and let (V (k) , W (k) ), U be independent,
U uniformly distributed on [0, 1] d . Let (V (k) , W (k) ) be optimal l 2 –
couplings of (µ, ν), i.e. V (k) D = µ, W (k) D = ν and l 2 2(µ, ν) = IE(V (k) − W (k) ) 2 .
Then using the independence properties and IEV (k) = IEW (k) = 0
⎛
⎞
2 d −1
l2(T 2 (µ), T (ν)) = l2
2 ∑
2
⎝ 〈U〉 k V (k) ∑
d −1
+ C(〈U〉), 〈U〉 k W (k) + C(〈U〉) ⎠
≤
=
k=0
⎛
⎞
2∑
d −1
IE ⎝ 〈U〉 k (V (k) − W (k) ) ⎠
2 d −1 ∑
k=0
k=0
IE [ 〈U〉 2 k(V (k) − W (k) ) 2]
= 2 d · IE〈U〉 2 0 · l 2 2(µ, ν)
k=0
2
=
( 2
3
) d
l 2 2(µ, ν) (32)
so T is a contraction on M 0,2 .
By Banach’s fixed point theorem T has a unique fixed point ρ in M 0,2 and
l 2 (T n (µ), ρ) → 0 (33)
exponentially fast for any µ ∈ M 0,2 .
We call a random variable X with distribution ρ also a fixed point of T .
(compare equation (25))
Theorem 3.3 (Limit theorem for the internal path length) The normalized
internal path length X n of a random quad tree converges w.r.t. l 2 to the
unique fixed point X in M 0,2 of the limiting operator T , i.e.
l 2 (X n , X) → 0. (34)
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