Edgeworth expansion for branching random walks and random trees Zakhar Kabluchko Westf¨ alische Wilhelms-Universit¨ at M¨ unster Joint work with Rudolf Gr¨ ubel Leibniz Universit¨ at Hannover AofA 2015, Strobl June 8, 2015
Branching random walk (BRW) Branching random walk models a random cloud of particles on Z . Random spatial motion of particles is combined with bran- ching. Definition of the BRW At time 0: One particle at 0. At time n : Every particle located, say, at x ∈ Z is replaced by a random cluster of N particles located at x + Z 1 , . . . , x + Z N . Here, � N k =1 δ ( Z k ) is a point process on Z . All random mechanisms are independent. 2
Profile of the branching random walk Consider a BRW on the lattice Z . Denote by L n ( k ) the number of particles located at site k ∈ Z at time n ∈ N 0 . Definition The random function k �→ L n ( k ) is called the profile of the BRW. 1 � 10 8 � � � � � � � 8 � 10 7 � � � � 6 � 10 7 � � � � 4 � 10 7 � � � � � 2 � 10 7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 20 � 10 0 10 20 3
Results Our aim is to obtain an asymptotic expansion of the profile as n → ∞ . As an application, we obtain a.s. limit theorems with non-degenerate limits for the occupation numbers L n ( k n ); the mode u n := arg max k ∈ Z L n ( k ); the height M n := max k ∈ Z L n ( k ) = L n ( u n ). In the setting of random trees these and related quantities were studied by Fuchs, Hwang, Neininger (2006), Chauvin, Drmo- ta, Jabbour-Hattab (2001), Katona (2005), Drmota, Hwang (2005), Devroye, Hwang (2006), Drmota, Janson, Neininger (2008). 4
Intensity Definition The intensity of the BRW at time n is the following measure on Z : ν n ( { k } ) := E L n ( k ) , k ∈ Z . Observation ν n is the n -th convolution power of ν 1 . Definition and assumption Let the cumulant generating function � e β k ν 1 ( { k } ) ϕ ( β ) := log k ∈ Z be finite for | β | < ε . 5
Biggins martingale Theorem (Uchiyama, 1982, Biggins, 1992) With probability 1, the martingale W n ( β ) := e − ϕ ( β ) n � L n ( k ) e β k k ∈ Z converges uniformly on | β | < ε to some random analytic func- tion W ∞ ( β ). Remark The random analytic function W ∞ encodes the “convolution difference” between the distribution of particles in the BRW at time n and the intensity measure ν n . 6
Local CLT for the BRW Theorem (Local form of the “Harris conjecture”) Let µ = ϕ ′ (0), σ 2 = ϕ ′′ (0) and x n ( k ) = k − µ n σ √ n , k ∈ Z . Then, with probability 1, � 1 L n ( k ) e ϕ (0) n = W ∞ (0) � n ( k ) + o e − 1 2 x 2 √ √ n , n → ∞ , 2 π n σ where the o -term is uniform in k ∈ Z . Remark The number of particles at time n is ≈ W ∞ (0) e ϕ (0) n . 7
Edgeworth expansion for the BRW Theorem (Gr¨ ubel, Kabluchko, 2015) Let µ = ϕ ′ (0), σ 2 = ϕ ′′ (0) and x n ( k ) = k − µ n σ √ n , k ∈ Z . Then, with probability 1 the following asymptotic expansion holds uniformly in k ∈ Z : L n ( k ) e ϕ (0) n ∼ W ∞ (0) � 1 + F 1 ( x n ( k )) + F 2 ( x n ( k )) � e − 1 2 x 2 n ( k ) √ √ n + . . . n 2 π n σ where � ϕ ′′′ (0) � 6 σ 3 ( x 3 − 3 x ) + W ′ ∞ (0) x F 1 ( x ) = , W ∞ (0) σ F 2 ( x ) = . . . . 8
Shift correction 1 � 10 8 � � � � � � � 8 � 10 7 � � � � 6 � 10 7 � � � � 4 � 10 7 � � � � � 2 � 10 7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 20 � 10 0 10 20 1 � 10 8 � � � � � � � 8 � 10 7 � � � � 6 � 10 7 � � � � 4 � 10 7 � � � � � 2 � 10 7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 20 � 10 0 10 20 9
Applications: The mode Edgeworth expansion can be applied to obtain a.s. limit theo- rems with non-degenerate limits for the occupation numbers L n ( k n ) the mode u n := arg max k ∈ Z L n ( k ) the height M n := max k ∈ Z L n ( k ) = L n ( u n ). Theorem (Gr¨ ubel, Kabluchko, 2015) There is a random variable N such that with probability 1, the mode at time n > N is equal to ⌊ u ∗ n ⌋ or ⌈ u ∗ n ⌉ , where n = ϕ ′ (0) n + W ′ W ∞ (0) − ϕ ′′′ (0) ∞ (0) u ∗ 2 σ 2 . 10
The mode Mode u n as a function of time n u n � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � n � � � � � � � � � � � � � � � � � � � � � � � 10 20 30 40 50 � 11
Applications: The height Theorem (Gr¨ ubel, Kabluchko, 2015) Let M n = max k ∈ Z L n ( k ) be the height of the BRW at time n . The a.s. subsequential limits of the sequence √ � � 2 π n σ M n ˜ M n := 2 σ 2 n 1 − W ∞ (0) e ϕ (0) n have the form (log W ∞ ) ′′ (0) + c , where c ∈ I and I ⊂ R is some compact set. The set I contains 1 element if ϕ ′ (0) is integer, contains finitely many elements if ϕ ′ (0) is rational, is an interval of length 1 / 4 is ϕ ′ (0) is irrational. 12
The height Normalized height ˜ M n as a function of time n M n � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � n � � � 0 50 100 150 200 M n � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � n � � � � 0 50 100 150 200 13
Applications: Occupation numbers Theorem (Gr¨ ubel, Kabluchko, 2015) Let k n = ⌊ ϕ ′ (0) n ⌋ + a , where a ∈ Z . The a.s. subsequential limits of the sequence √ 2 πσ 3 n 3 / 2 e − ϕ (0) n ( L n ( k n ) − W ∞ (0) E L n ( k n )) have the form � c + ϕ ′′′ (0) � − 1 W ′ 2 W ′′ ∞ (0) ∞ (0) , 2 σ 2 where c ∈ J and J ⊂ R is some compact set which can be described explicitly. 14
Recommend
More recommend