On the Internal Path Length of dâdimensional Quad Trees

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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
Corollary 4.2 Var (Y n ) ∼ v d n 2 (47) with v d given in (44). In the Quicksort case d = 1 Rösler [16] showed finiteness of the Laplace transform of X and convergence of the Laplace transforms of X n to that of X. In particular this implies finiteness and convergence of higher order moments. Röslers proof directly extends to the case d ≥ 1. Lemma 4.1 in Rösler [16] holds in any dimension as follows. Lemma 4.3 ∀ L > 0 : ∃ K L > 0 : ∀ n ∈ IN and ∀ λ ∈ [−L, L] holds IE exp(λX n ) ≤ exp(λ 2 K L ). (48) Proof: In place of the random variable U n in Röslers proof use V n := ‖I (n) /n‖ 2 − 1. Then a) − 1 ≤ V n < 0 for all n ∈ IN b) sup IEV n < 0 n∈IN c) sup ‖C n ‖ ∞ < ∞ by Lemma 3.1. n∈IN For the proof of b) note that IEV n < 0 for all n ∈ IN and V n D → ‖〈U〉‖ 2 − 1 which implies by boundedness of V n , IEV n → IE(‖〈U〉‖ 2 −1) < 0. From a)–c) one obtains (48) as in Rösler [16]. Theorem 4.4 (Convergence of Laplace transforms) For the normalized internal path length X n holds IE exp(λX n ) → IE exp(λX), λ ∈ IR 1 . (49) Proof: The exponential bound in (48) implies uniform integrability of exp(λX n ) which by Theorem 3.3 yields (49). Finally using the expansion of the mean IEY n in (3) one obtains as in Corollary 4.3 of Rösler [16] the following bounds for (large) deviations. 15
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 and 12: D Proof: Let X n (k) = X n , X (k)
Page 13: n−1 +2 d ∑ i=n 0 IP ⎛ ⎝ I(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

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On the Internal Path Length of dâdimensional Quad Trees