On the Internal Path Length of dâdimensional Quad Trees

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Now observe that 2α := min z k > 0 0≤k≤2 d −1 since z ∈ (0, 1] 2d . This implies z (n) k ≥ αn for 0 ≤ k ≤ 2 d −1 and n sufficiently large. Let ¯R n := sup i≥n |R i |. Then ( ¯R n ) n∈IN is decreasing and ¯R n /n → 0. For n sufficiently large it follows 2 1 ∑ d −1 R (n) ∣n z k k=0 ∣ ≤ ≤ 1 n 1 n 2 d −1 ∑ k=0 2 d −1 ∑ k=0 ¯R z (n) k ¯R ⌊αn⌋ ≤ 2 d α 1 ⌊αn⌋ ¯R ⌊αn⌋ → 0 for n → ∞. (23) The third summand of (22) also tends to zero. So (22) implies C n (z (n) ) = C(z (n) /n) + o(1). Therefore, continuity of C and the triangle inequality imply ∣ ∣C n (z (n) ) − C(z) ∣ ∣ ∣ ≤ ∣∣Cn (z (n) ) − C(z (n) /n) ∣ ∣ ∣ + ∣∣C(z (n) /n) − C(z) ∣ ∣ −→ 0. In order to get an estimate for C n (z (n) ) uniformly in z (n) ∈ G n let Then (22) implies The second claim follows. L := sup |R n /n| < ∞. n∈IN |C n (z (n) )| ≤ |C(z (n) /n)| + 2 d L + o(1) ≤ ‖C‖ ∞ + 2 d L + o(1). (24) Lemmas 2.1, 3.1 suggest that a limit X of (X n ) is a solution of the limiting equation X D = 2∑ d −1 k=0 〈U〉 k X (k) + C(〈U〉) (25) 8
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 . Define M 0,2 := { µ ∈ M 1 (IR 1 , B 1 ) | IEµ = 0, Var µ < ∞} (26) where IEµ, Var µ are defined as expectation respectively variance of a corresponding random variable and M 1 (IR 1 , B 1 ) denotes the space of probability measures on the real line. Define the random affine operator T : M 1 (IR 1 , B 1 ) → M 1 (IR 1 , B 1 ), T (µ) D = 2∑ d −1 k=0 〈U〉 k Z (k) + C(〈U〉) (27) where (Z (k) ), U are independent, Z (k) D = µ and U is uniformly distributed on [0, 1] d . Our aim is to show that T is the limiting operator of the recursive sequence (X n ) in (14). Supply M 0,2 ⊂ M 1 (IR 1 , B 1 ) with the minimal l 2 –metric l 2 (µ, ν) = inf{(IE|X − Y | 2 ) 1/2 : X D = µ, Y D = ν}. (28) For random variables X, Y we use synonymously l 2 (X, Y ) = l 2 (IP X , IP Y ). Then (M 0,2 , l 2 ) is a complete metric space and l 2 (µ n , µ) → 0 is equivalent to ∫ ∫ µ D n → µ and x 2 dµ n (x) → x 2 dµ(x). (29) (see Rachev [13]) Lemma 3.2 T : M 0,2 → M 0,2 is a contraction w.r.t. l 2 : ( ) 2 d/2 l 2 (T (µ), T (ν)) ≤ l 2 (µ, ν) for all µ, ν ∈ M 0,2 . (30) 3 Proof: Obviously Var (T (µ)) < ∞ and IET (µ) = 1 + 2 d 2 d −1 ∑ k=0 IE [〈U〉 k ln〈U〉 k ] = 1 + 2 d 2d IE [〈U〉 0 ln〈U〉 0 ] = 1 + 2 d 2d ∫[0,1] d u 1 · · · u d ln(u 1 · · · u d ) dλ d (u) 9
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: where x ln x is defined to be 0 for
Page 11 and 12: D Proof: Let X n (k) = X n , X (k)
Page 13 and 14: n−1 +2 d ∑ i=n 0 IP ⎛ ⎝ I(n
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

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On the Internal Path Length of dâdimensional Quad Trees