Normal distribution in the subanalytic setting Julia Ruppert - PowerPoint PPT Presentation

Normal distribution in the subanalytic setting Julia Ruppert University of Passau Faculty of Informatics and Mathematics July 2015

Outline 1. Motivation 2. Statement of the problem 3. Results 4. Summary University of Passau Julia Ruppert 1 / 13

Motivation Parameterized integrals in the subanalytic setting: ◮ Comte, Lion, Rolin: � f ( x , y ) dy ◮ Cluckers, Miller: f ( x , y )( log ( g ( x , y ))) n dy � ◮ Cluckers, Comte, Miller, Rolin, Servi: � e i g ( x , y ) f ( x , y ) dy − y 2 ◮ Now: � 2 t f ( x , y ) dy e University of Passau Julia Ruppert 2 / 13

Motivation Parameterized integrals in the subanalytic setting: ◮ Comte, Lion, Rolin: � f ( x , y ) dy ◮ Cluckers, Miller: f ( x , y )( log ( g ( x , y ))) n dy � ◮ Cluckers, Comte, Miller, Rolin, Servi: � e i g ( x , y ) f ( x , y ) dy − y 2 ◮ Now: � 2 t f ( x , y ) dy e University of Passau Julia Ruppert 2 / 13

Motivation Parameterized integrals in the subanalytic setting: ◮ Comte, Lion, Rolin: � f ( x , y ) dy ◮ Cluckers, Miller: f ( x , y )( log ( g ( x , y ))) n dy � ◮ Cluckers, Comte, Miller, Rolin, Servi: � e i g ( x , y ) f ( x , y ) dy − y 2 ◮ Now: � 2 t f ( x , y ) dy e University of Passau Julia Ruppert 2 / 13

Motivation Parameterized integrals in the subanalytic setting: ◮ Comte, Lion, Rolin: � f ( x , y ) dy ◮ Cluckers, Miller: f ( x , y )( log ( g ( x , y ))) n dy � ◮ Cluckers, Comte, Miller, Rolin, Servi: � e i g ( x , y ) f ( x , y ) dy − y 2 ◮ Now: � 2 t f ( x , y ) dy e University of Passau Julia Ruppert 2 / 13

Brownian Motion Definition (Brownian Motion in R ) An one dimensional stochastic process ( B t ) t ≥ 0 is called Brownian Motion in R with start value z if it is characterised by the following facts: ◮ B 0 = z ◮ Let 0 ≤ s < t. Then B t − B s is normally distributed with expected value 0 and variance t − s. ◮ Let n ≥ 1 , 0 ≤ t 0 < t 1 < . . . < t n .Then B t 0 , B t 1 − B t 0 , . . . , B t n − B t n − 1 are independent random variables. ◮ Every path is almost surely continuous. University of Passau Julia Ruppert 3 / 13

Brownian Motion Definition (Brownian Motion in R ) An one dimensional stochastic process ( B t ) t ≥ 0 is called Brownian Motion in R with start value z if it is characterised by the following facts: ◮ B 0 = z ◮ Let 0 ≤ s < t. Then B t − B s is normally distributed with expected value 0 and variance t − s. ◮ Let n ≥ 1 , 0 ≤ t 0 < t 1 < . . . < t n .Then B t 0 , B t 1 − B t 0 , . . . , B t n − B t n − 1 are independent random variables. ◮ Every path is almost surely continuous. University of Passau Julia Ruppert 3 / 13

Brownian Motion Definition (Brownian Motion in R ) An one dimensional stochastic process ( B t ) t ≥ 0 is called Brownian Motion in R with start value z if it is characterised by the following facts: ◮ B 0 = z ◮ Let 0 ≤ s < t. Then B t − B s is normally distributed with expected value 0 and variance t − s. ◮ Let n ≥ 1 , 0 ≤ t 0 < t 1 < . . . < t n .Then B t 0 , B t 1 − B t 0 , . . . , B t n − B t n − 1 are independent random variables. ◮ Every path is almost surely continuous. University of Passau Julia Ruppert 3 / 13

Brownian Motion Definition (Brownian Motion in R ) An one dimensional stochastic process ( B t ) t ≥ 0 is called Brownian Motion in R with start value z if it is characterised by the following facts: ◮ B 0 = z ◮ Let 0 ≤ s < t. Then B t − B s is normally distributed with expected value 0 and variance t − s. ◮ Let n ≥ 1 , 0 ≤ t 0 < t 1 < . . . < t n .Then B t 0 , B t 1 − B t 0 , . . . , B t n − B t n − 1 are independent random variables. ◮ Every path is almost surely continuous. University of Passau Julia Ruppert 3 / 13

Brownian Motion Definition (Brownian Motion in R n ) An n-dimensional stochastic process ( B t = ( B 1 t , . . . , B n t )) t ≥ 0 is called Brownian Motion in R n with start value z ∈ R n if every stochastic process ( B i t ) t ≥ 0 is a Brownian Motion in R for i ∈ { 1 , . . . , n } , the stochastic processes B 1 t , . . . , B n t are independent for every t ≥ 0 and B 0 = z. University of Passau Julia Ruppert 4 / 13

Brownian Motion Let A ⊂ R n be a borel set and let z ∈ R n be the start value. The probability for B t ∈ A at time t is given by  δ z ( A ) , t = 0 ,     P ( B t ∈ A ) = � e − | x − z | 2 1 dx , t > 0 . 2 t n   (2 π t ) 2   A where  1 , z ∈ A ,  δ z ( A ) = 0 , z / ∈ A .  University of Passau Julia Ruppert 5 / 13

Brownian Motion Let A ⊂ R n be a borel set and let z ∈ R n be the start value. The probability for B t ∈ A at time t is given by  δ z ( A ) , t = 0 ,     P ( B t ∈ A ) = � e − | x − z | 2 1 dx , t > 0 . 2 t n   (2 π t ) 2   A where  1 , z ∈ A ,  δ z ( A ) = 0 , z / ∈ A .  University of Passau Julia Ruppert 5 / 13

Brownian Motion Let A ⊂ R n be a borel set and let z ∈ R n be the start value. The probability for B t ∈ A at time t is given by  δ z ( A ) , t = 0 ,     P ( B t ∈ A ) = � e − | x − z | 2 1 dx , t > 0 . 2 t n   (2 π t ) 2   A where  1 , z ∈ A ,  δ z ( A ) = 0 , z / ∈ A .  University of Passau Julia Ruppert 5 / 13

Motivation 6 4 2 A 0 −2 −4 0 5 10 15 20 ◮ microscopic: wild ◮ macroscopic: tame if A is tame ?! University of Passau Julia Ruppert 6 / 13

Statement of the problem Let A ⊂ R n × R m be a semialgebraic set. Let f : R n × R m × R ≥ 0 − → [0 , 1]  δ z ( A a ) , t = 0 ,   −| x − z | 2 ( a , z , t ) �→ 1 � e dx , t > 0 . 2 t n  (2 π t ) 2  A a Question: ◮ Definability? ◮ Asymptotics? University of Passau Julia Ruppert 7 / 13

Results for R Let A ⊂ R n × R be definable in an o-minimal structure M . The function f : R n × R × R ≥ 0 − → [0 , 1]  δ z ( A a ) , t = 0   ( a , z , t ) �→ P z ( B t ∈ A a ) = − ( x − z )2 1 � e dx , t > 0 √ 2 t 2 π t   A a is definable in an expansion of the o-minimal structure M . University of Passau Julia Ruppert 8 / 13

Results for R Let A ⊂ R n × R be definable in an o-minimal structure M . The function f : R n × R × R ≥ 0 − → [0 , 1]  δ z ( A a ) , t = 0   ( a , z , t ) �→ P z ( B t ∈ A a ) = − ( x − z )2 1 � e dx , t > 0 √ 2 t 2 π t   A a is definable in an expansion of the o-minimal structure M . University of Passau Julia Ruppert 8 / 13

β ( a ) � − ( x − z )2 � − ( x − z )2 e dx = e dx 2 t 2 t C a α ( a ) √ π √ � � β ( a ) − z � � α ( a ) − z �� = 2 t erf √ − erf √ 2 2 t 2 t University of Passau Julia Ruppert 9 / 13

β ( a ) � − ( x − z )2 � − ( x − z )2 e dx = e dx 2 t 2 t C a α ( a ) √ π √ � � β ( a ) − z � � α ( a ) − z �� = 2 t erf √ − erf √ 2 2 t 2 t University of Passau Julia Ruppert 9 / 13

β ( a ) � − ( x − z )2 � − ( x − z )2 e dx = e dx 2 t 2 t C a α ( a ) √ π √ � � β ( a ) − z � � α ( a ) − z �� = 2 t erf √ − erf √ 2 2 t 2 t Theorem (Speissegger) Suppose that I ⊆ R is an open interval, a ∈ I and g : I → R is definable in the Pfaffian closure P ( M ) and continuous.Then its antiderivative x � F : I → R given by F ( x ) := g ( t ) dt is also definable in P ( M ) . a University of Passau Julia Ruppert 9 / 13

β ( a ) � − ( x − z )2 � − ( x − z )2 e dx = e dx 2 t 2 t C a α ( a ) √ π √ � � β ( a ) − z � � α ( a ) − z �� = 2 t erf √ − erf √ 2 2 t 2 t By Speissegger f ( a , z , t ) is definable in P ( M ). University of Passau Julia Ruppert 9 / 13

Results for higher dimensions Let A ⊂ R n × R 2 be a globally subanalytic set. We assume that A a is uniformly bounded and the start value z is 0. Then the function f : R n × R ≥ 0 − → [0 , 1] 1 � −| x | 2 2 t dx , t > 0 ( a , t ) �→ P 0 ( B t ∈ A a ) = e 2 π t A a a) is definable in R an for t → ∞ (by Comte, Lion, Rolin). b) For t → 0 we can establish an asymptotic expansion ∞ n � 2 q , f ∼ d n ( a ) t n =0 where d n ( a ) is globally subanalytic for all n ∈ N 0 . University of Passau Julia Ruppert 10 / 13

Sketch of the proof in the case without parameters: With polar coordinate transformation and cell decomposition ψ ( r ) β 1 1 � e − r 2 � � e − r 2 2 t r d ( r , ϕ ) 2 t r d ϕ dr = 2 π t 2 π t r = α C ϕ = η ( r ) β ϕ 1 � e − r 2 2 t r ψ ( r ) dr = 2 π t ψ ( r ) α C β − 1 � e − r 2 η ( r ) 2 t r η ( r ) dr 2 π t α r ] [ University of Passau Julia Ruppert 11 / 13

Sketch of the proof in the case without parameters: With polar coordinate transformation and cell decomposition ψ ( r ) β 1 1 � e − r 2 � � e − r 2 2 t r d ( r , ϕ ) 2 t r d ϕ dr = 2 π t 2 π t r = α C ϕ = η ( r ) β ϕ 1 � e − r 2 2 t r ψ ( r ) dr = 2 π t ψ ( r ) α C β − 1 � e − r 2 η ( r ) 2 t r η ( r ) dr 2 π t α r ] [ University of Passau Julia Ruppert 11 / 13