General Edgeworth expansions Examples General Edgeworth expansions with One-split branching random walks applications to profiles of random trees Random trees Mode and width Alexander Marynych Taras Shevchenko National University of Kyiv and University of M¨ unster joint work with Z. Kabluchko (M¨ unster, Germany) and H. Sulzbach (Birmingham, UK) Alexander Marynych, Kyiv General Edgeworth expansions 1/25
Assumptions on profiles General Edgeworth expansions Random profiles A random profile is an arbitrary sequence Bell and Hermite polynomials The main theorem L n = ( L n ( k )) k ∈ Z , n ∈ N Examples One-split branching random walks of real-valued stochastic processes on Z . Random trees Mode and width Alexander Marynych, Kyiv General Edgeworth expansions 2/25
Assumptions on profiles General Edgeworth expansions Random profiles A random profile is an arbitrary sequence Bell and Hermite polynomials The main theorem L n = ( L n ( k )) k ∈ Z , n ∈ N Examples One-split branching random walks of real-valued stochastic processes on Z . Random trees Assumption A1: There exists an open interval ( β − , β + ) ⊂ R Mode and width containing zero such that for every n ∈ N and every β ∈ ( β − , β + ) , | L n ( k ) | e βk < ∞ � a.s. k ∈ Z Alexander Marynych, Kyiv General Edgeworth expansions 2/25
Assumptions on profiles General Edgeworth expansions Random profiles A random profile is an arbitrary sequence Bell and Hermite polynomials The main theorem L n = ( L n ( k )) k ∈ Z , n ∈ N Examples One-split branching random walks of real-valued stochastic processes on Z . Random trees Assumption A1: There exists an open interval ( β − , β + ) ⊂ R Mode and width containing zero such that for every n ∈ N and every β ∈ ( β − , β + ) , | L n ( k ) | e βk < ∞ � a.s. k ∈ Z The interval ( β − , β + ) may be infinite. For example, if for every n ∈ N the support of a profile { k ∈ Z : L n ( k ) � = 0 } is finite with probability one, then Assumption A1 holds with ( β − , β + ) = R . Alexander Marynych, Kyiv General Edgeworth expansions 2/25
Assumptions on profiles General Edgeworth expansions Random profiles Bell and Hermite polynomials Assume that there exist The main theorem Examples a sequence ( w n ) n ∈ N ⊂ R such that lim n →∞ w n = + ∞ ; One-split branching random an open domain D ⊂ { β ∈ C : β − < Re β < β + } such that walks D ∩ R = ( β − , β + ) ; Random trees a deterministic analytic function ϕ : D → C such that for every Mode and width β ∈ ( β − , β + ) we have ϕ ( β ) ∈ R and ϕ ′′ ( β ) > 0 . From Assumption A1 it follows that with probability one, the normalized Laplace transform W n ( β ) := e − ϕ ( β ) w n � L n ( k ) e βk , β ∈ D , k ∈ Z is a random analytic function on D for every n ∈ N . Alexander Marynych, Kyiv General Edgeworth expansions 3/25
Assumptions on profiles General Edgeworth expansions Random profiles Bell and Hermite polynomials The main theorem Assumption A2: With probability one the sequence of random Examples analytic functions ( W n ) n ∈ N converges locally uniformly on D , as n → ∞ , to a random analytic function W ∞ such that One-split branching random walks P [ W ( β ) � = 0 for all β ∈ ( β − , β + )] = 1 . Random trees Mode and width Alexander Marynych, Kyiv General Edgeworth expansions 4/25
Assumptions on profiles General Edgeworth expansions Random profiles Bell and Hermite polynomials The main theorem Assumption A2: With probability one the sequence of random Examples analytic functions ( W n ) n ∈ N converges locally uniformly on D , as n → ∞ , to a random analytic function W ∞ such that One-split branching random walks P [ W ( β ) � = 0 for all β ∈ ( β − , β + )] = 1 . Random trees Assume also that the speed of convergence is superpolynomial in w n : Mode and width Assumption A3: For every compact set K ⊂ D and every r ∈ N there exists an a.s. finite C K,r such that for all n ∈ N , | W n ( β ) − W ∞ ( β ) | < C K,r w − r sup n . β ∈ K Alexander Marynych, Kyiv General Edgeworth expansions 4/25
Assumption on profiles General Edgeworth expansions Random profiles Bell and Hermite polynomials The main theorem Examples The next assumption is technical. In the classical Edgeworth expansion for sums of i.i.d. variables it corresponds to the assumption One-split branching random that Z is a minimal lattice containing the support of the step. walks Random trees Mode and width Alexander Marynych, Kyiv General Edgeworth expansions 5/25
Assumption on profiles General Edgeworth expansions Random profiles Bell and Hermite polynomials The main theorem Examples The next assumption is technical. In the classical Edgeworth expansion for sums of i.i.d. variables it corresponds to the assumption One-split branching random that Z is a minimal lattice containing the support of the step. walks Random trees Assumption A4: For every compact set K ⊂ ( β − , β + ) , every a > 0 Mode and width and r ∈ N 0 , we have � � � π � � e − ϕ ( β ) w n � L n ( k ) e k ( β + iu ) = o ( w − r � � sup d u n ) a.s. � � β ∈ K a � � k ∈ Z � � as n → ∞ . Alexander Marynych, Kyiv General Edgeworth expansions 5/25
Bell polynomials General Edgeworth The complete Bell polynomials B j ( z 1 , . . . , z j ) are defined by the expansions formal identity Random profiles Bell and Hermite polynomials The main theorem ∞ ∞ x j x j Examples � � exp j ! z j = j ! B j ( z 1 , . . . , z j ) . One-split branching random j =1 j =0 walks Random trees Therefore, B 0 = 1 and for j ∈ N , Mode and width � z j � z 1 � i j j ! � i 1 . . . � ′ B j ( z 1 , . . . , z j ) = , i 1 ! . . . i j ! 1! j ! where the sum � ′ is taken over all i 1 , . . . , i j ∈ N 0 such that 1 i 1 + 2 i 2 + . . . + ji j = j . Alexander Marynych, Kyiv General Edgeworth expansions 6/25
Bell polynomials General Edgeworth The complete Bell polynomials B j ( z 1 , . . . , z j ) are defined by the expansions formal identity Random profiles Bell and Hermite polynomials The main theorem ∞ ∞ x j x j Examples � � exp j ! z j = j ! B j ( z 1 , . . . , z j ) . One-split branching random j =1 j =0 walks Random trees Therefore, B 0 = 1 and for j ∈ N , Mode and width � z j � z 1 � i j j ! � i 1 . . . � ′ B j ( z 1 , . . . , z j ) = , i 1 ! . . . i j ! 1! j ! where the sum � ′ is taken over all i 1 , . . . , i j ∈ N 0 such that 1 i 1 + 2 i 2 + . . . + ji j = j . The first few Bell polynomials: B 2 ( z 1 , z 2 ) = z 2 B 3 ( z 1 , z 2 , z 3 ) = z 3 B 1 ( z 1 ) = z 1 , 1 + z 2 1 +3 z 1 z 2 + z 3 . Alexander Marynych, Kyiv General Edgeworth expansions 6/25
Hermite polynomials General Edgeworth expansions Random profiles Bell and Hermite polynomials The main theorem Examples The “probabilist” Hermite polynomial He n ( x ) of degree n is defined One-split branching random by walks � n 2 x 2 � − d 1 e − 1 2 x 2 . He n ( x ) = e Random trees d x Mode and width The first few Hermite polynomials are: He 2 ( x ) = x 2 − 1 , He 3 ( x ) = x 3 − 3 x, He 1 ( x ) = x, He 4 ( x ) = x 4 − 6 x 2 + 3 , He 6 ( x ) = x 6 − 15 x 4 + 45 x 2 − 15 . Alexander Marynych, Kyiv General Edgeworth expansions 7/25
The main theorem General Edgeworth expansions Consider a random profile L 1 , L 2 , . . . which satisfies assumptions Random profiles A1–A4. Bell and Hermite polynomials The main theorem Consider the tilted profile k �→ e βk − ϕ ( β ) w n L n ( k ) and define its Examples “mean” µ ( β ) and the “standard deviation” σ ( β ) : One-split branching random walks µ ( β ) = ϕ ′ ( β ) , σ 2 ( β ) = ϕ ′′ ( β ) , Random trees and “the normalized coordinates” Mode and width x n ( k ) = x n ( k ; β ) = k − µ ( β ) w n σ ( β ) √ w n , k ∈ Z . Alexander Marynych, Kyiv General Edgeworth expansions 8/25
The main theorem General Edgeworth expansions Consider a random profile L 1 , L 2 , . . . which satisfies assumptions Random profiles A1–A4. Bell and Hermite polynomials The main theorem Consider the tilted profile k �→ e βk − ϕ ( β ) w n L n ( k ) and define its Examples “mean” µ ( β ) and the “standard deviation” σ ( β ) : One-split branching random walks µ ( β ) = ϕ ′ ( β ) , σ 2 ( β ) = ϕ ′′ ( β ) , Random trees and “the normalized coordinates” Mode and width x n ( k ) = x n ( k ; β ) = k − µ ( β ) w n σ ( β ) √ w n , k ∈ Z . Define the “deterministic cumulants” κ j ( β ) and the “random cumulants” χ j ( β ) as follows: κ j ( β ) = ϕ ( j ) ( β ) , χ j ( β ) = (log W ∞ ) ( j ) ( β ) . Alexander Marynych, Kyiv General Edgeworth expansions 8/25
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