On the Internal Path Length of dâdimensional Quad Trees

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+2 · IE ⎡ ⎣ I(n) k n ( X (k) I (n) k ) ⎛ − X (k) ⎝ I(n) k n ⎞ ⎤ − 〈U〉 ⎠ k X (k) ⎦ . (38) By independence and (36) the second term in (38) converges to zero. With the Cauchy-Schwarz-inequality the third term in it’s absolute value is estimated from above by ⎢ 2IE ⎣ = o(1)IE ⎡⎛ ⎞2 ⎛ ⎝ I(n) k ⎠ ⎝ I(n) k n n ⎞ ⎤ 2 (X − 〈U〉 ⎠ ) (k) 2 k ⎥ ⎦ IE ( ) 2 X (k) − X (k) I (n) k ( ) 2 X (k) − X (k) , (39) I (n) k where again (36) has been used. With (35)–(39) and denoting a n := l2(X 2 n , X) we derive a n ≤ 2 d IE n−1 = 2 d ∑ i=0 ( ⎛ ⎝ I(n) k n ⎞ ( ) ) 2 + o(1) ⎠ X (k) − X (k) + b I (n) n k IP ({(I (n) /n) = (i/n)}) k ×((i/n) 2 + o(1)) IE(X (k) i − X (k) ) 2 + b n (40) where b n → 0 for n → ∞. From Corollary 2.2 we conclude n−1 a n ≤ 2 d ∑ i=1 IP ({(I (n) k /n) = (i/n)}) ((i/n)2 + o(1)) sup a i + b n 1≤i≤n−1 = ((2/3) d + o(1)) sup a i + b n (41) 1≤i≤n−1 which implies that (a n ) is bounded. This implies as in Rösler [16] that for a given ɛ > 0 there exists n 0 such that for n ≥ n 0 a n ≤ a + ɛ with a := lim sup a n n→∞ and the prefactor in (41) is uniformly less than a γ < 1. Therefore ⎛ ⎞ n 0 −1 a n ≤ 2 d ∑ IP ⎝ I(n) k n = i ( ( ) ⎠ i 2 + o(1)) a i n n i=1 12
n−1 +2 d ∑ i=n 0 IP ⎛ ⎝ I(n) k n ⎞ = i ( ( ) ⎠ i 2 + o(1)) (a + ɛ) + b n n n ≤ γ(a + ɛ) + o(1). (42) Then 0 ≤ a = lim sup a n ≤ γ(a + ɛ) which implies a = 0. 4 Moments and tail of the internal path length Let X be the unique solution of the fixed point equation (25) for the internal path length in M 0,2 X D = 2∑ d −1 k=0 〈U〉 k X (k) + C(〈U〉) (43) where X (k) are iid copies of X and {X (k) , k = 0, . . . 2 d −1}, U are independent, U uniformly distributed on [0, 1] d . Then (43) implies recursive equations for the moments IEX k of X which can be solved (in principle) as soon as we know the existence of higher order moments. For the variance of X we obtain (note that IEX = 0) Proposition 4.1 (Variance of X) The variance of the limit X of the normalized internal path length of a d–dimensional random quad tree is given by In particular v d = 21 − 2π 2 9d(1 − (2/3) d ) . (44) v 1 = 0.4202 . . . , v 2 = 0.1260 . . . , v 3 = 0.0663 . . . Proof: (43) and the independence properties imply ( ( ) 2 d ) −1 IEX 2 = 1 − IEC 2 (〈U〉). 3 13
Page 1 and 2: On the Internal Path Length of d-di
Page 3 and 4: transform; X has exponential tails.
Page 5 and 6: when either two random variables ar
Page 7 and 8: where x ln x is defined to be 0 for
Page 9 and 10: where X (k) are iid copies of X and
Page 11: D Proof: Let X n (k) = X n , X (k)
Page 15 and 16: Corollary 4.2 Var (Y n ) ∼ v d n
Page 17 and 18: the question arises under which con
Page 19 and 20: (b) exponential moments exist and c
Page 21 and 22: [5] Devroye, L. & Laforest, L. 1990

On the Internal Path Length of dâdimensional Quad Trees

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On the Internal Path Length of dâdimensional Quad Trees