L evy Computability of Probability Distribution Functions Takakazu - PowerPoint PPT Presentation

< 13Nancy > 1 L´ evy Computability of Probability Distribution Functions Takakazu Mori Yoshiki Tsujii Mariko Yasugi 森 隆一 辻井 芳樹 八杉 満利子 Kyoto-Sangyo University ~ }

< 13Nancy > 2 0. Introduction vague conv. ★ ✥ R e itx µ ( dx ) � set of all probability ■ ❅ ★ ✥ ✘ measures on R unif. conv. ✘ P ❅ ✘ ✾ ✘ ✘ ✧ ✦ ❅ ✿ ✘ ✘✘✘✘ ❅ ❅ CF ❅ ✧ ✦ ❅ Bochner ❅ ✻ ❅ ❅ ❅ positive definite ❅ ❅ uniformly continuous F ( x ) = µ (( a, b ]) = ❘ ❅ ❅ ❅ ■ ’ (0) = 1 ★ ✥ ❅ unif. conv. µ (( −∞ , x ]) F ( b ) − F ( a ) ❅ ❅ ❄ ❅ ❅ Inversion ❅ Fine case ★ ✥ ❅ C L ❅ Formula ✧ ✦ ✘ ✿ ❘ ❅ ❅ ✘✘ ✘ ✘ ✾ ✘ F L´ evy ✧ ✦ increasing Lipschitz set of all probability f ( `1 ) = 0 ; f (+ 1 ) = 1 distribution functions on R pointwase conv. Discontinuous at points of continuity Figure 1: The spaces P , F , C L Notations: µ ↔ F ↔ L F = f (unless oterwise stated) ~ }

< 13Nancy > 3 0 Introduction 1 Classical Theory 2 L´ evy height and L´ evy metric 3 Computability on P 4 L´ evy computabilities 5 L´ evy Computability and Fine Computability for Probability Distribution Functions 6 Effective L´ evy Convergence ~ }

< 13Nancy > 4 Computabilities on F We had treated Fine computability and effective Fine convergence We seek: computability and effective convergence on F relations to Fine computabilities on F relations to computability and effective convergence on P Translated Dirac measure δ 1 3 is computable, but the corresponding probability distribution functions is not Fine computable. We have an example of probability distribution function F such that F (0) is not computable, but the corresponding µ is computable (Example 4.2) We propose L´ evy computability and effective L´ evy convergence on F . ~

< 13Nancy > 5 Convergence on P motivation convergence ↔ determining class ↔ computability a class A of functions or of sets is a determining class if µ ( η ) = ν ( η ) for ∀ η ∈ A implies µ = ν ~ }

< 13Nancy > 6 1. Classical Theory about probability measures on R (Summary) Probability Distribution Functions are characterized by (F-i) monotonically increasing (F-ii) right-continuous (F-iii) F ( ∞ ) = 1, F ( −∞ ) = 0 Notations: F : the set of all probability distribution functions C κ : the set of all continuous functions with compact support C b : the set of all bounded continuous functions � µ ( f ) = R f ( x ) µ ( dx ) { g ℓ } : a dense sequence of C κ ∞ � 2 − ℓ ( | µ ( g ℓ ) − ν ( g ℓ ) | ∧ 1) { g ℓ } -metric : d { g ℓ } ( µ, ν ) = ℓ =1 ~ }

< 13Nancy > 7 The following converences are equivalent ([1], [6]). (i) µ m ( f ) → µ ( f ) ∀ f ∈ C κ (vague) (ii) µ m ( f ) → µ ( f ) ∀ f ∈ C b (weak) (iii) lim inf m µ m ( G ) ≥ µ ( G ) ∀ G : open domain theory S2007[23] (iv) d { g ℓ } ( µ m , µ ) → 0 { g ℓ } : dense in C κ W1999[27][0,1] [0 , 1] (iii ′ ) lim inf m µ m ( I ) ≥ µ ( I ) ∀ I : open interval W1999[27] Intervals SS2005[22] R (v) ϕ m ( t ) → ϕ ( t ) for any t (vi) F m ( x ) → F ( x ) ∀ x a point of continuity of F ⇔ µ ( { x } ) = 0, { ( −∞ , x ] } (vii) d L ( F m , F ) → 0 L´ evy (other metrices: cf. [2]) ~ }

< 13Nancy > 8 2. L´ evy height and L´ evy metric L´ evy [11], Ito [6] Notations: G F = { ( x, y ) | F − ( x ) ≤ y ≤ F ( x )) , x ∈ R } ( F − ( x ) = lim y ↑ x F ( y ) ) ∀ t ∈ R , ( x ( t ) , y ( t )) = the unique crossing point of X + Y = t with G F Definition 2.1 L´ evy height: L F ( t ) = y ( t ) � � Definition 2.2 L´ evy metric (distance): d L ( F, G ) = sup � L F ( t ) − L G ( t ) � t ∈ R Definition 2.3 L´ evy convergence: d L ( F n , F ) → 0 Remark 2.4 d L ( F, G ) is equal to the L´ evy(-Prokhorov) metric, that is, d L ( F, G ) = inf { ǫ > 0 | F ( x − ǫ ) − ǫ ≤ G ( x ) ≤ F ( x + ǫ ) + ǫ , for all x } ~ }

< 13Nancy > 9 F f ( t ) = L F ( t ) ˜ ˜ a b x F ` ( x ) t F ( x ) = t ` f ( t ) L F D x ˜ ˜ F ` ( b ) ˜ ˜ ˜ F ` ( x ) t F ( x ) F ( a ) F ( b ) Figure 2: F and L F Put L F ( t ) = f ( t ). D x = { t | x = t − f ( t ) } sup { f ( t ) | t ∈ D x } − inf { f ( t ) | t ∈ D x } = the jump of F at x ~

< 13Nancy > 10 Properties of L´ evy height Let C L be the set of functions which satisfy the following: (Li) 0 ≤ f ( t ) − f ( s ) ≤ t − s if s ≤ t . (Lii) lim t →−∞ f ( t ) = 0. (Liii) lim t →∞ f ( t ) = 1. Proposition 2.5 L : F → C L one-to-one and onto C L : closed convex subset of C b w.r.t. the sup-norm (distance) d ∞ Hence, ( C L , d ∞ ) is complete ~ }

< 13Nancy > 11 Properties of ˜ F , ˜ D x = [ ˜ F − ( x ) , ˜ F − , L F F ( x )] ˜ • F ( x ) = x + F ( x ) is strictly increasing and continuous L F ( t ) = F ( ˜ F − 1 ( t )) • [ ˜ F − ( x ) , ˜ F ( x )) ∩ Range ( F ) = φ ˜ F − 1 ( t ) is computable if F is computable • • F ( x ) = f (sup D x ) ˆ • f ( t ) = t − f ( t ) is a nondecreasing continuous function from R onto R

< 13Nancy > 12 Example 2.6 Dirac measures δ a , D a : prof. dist. function • D a ( x ) = 0 if x < a and = 1 if x ≥ a  0 if t ≤ a   • L D a ( t ) = t − a if a ≤ t ≤ 1 + a  1 if t ≥ 1 + a  • d L ( D a , D b ) = d ∞ ( L D a , L D b ) = | a − b | ∧ 1

< 13Nancy > 13 3. Computability on P We employ computable notions on R by Pour-El and Richards. Definition 3.1 { µ m } is computable ⇌ { µ m ( f n ) } is computable for any computable sequence f n with compact support ( L ( n ) s.t. f n ( x ) = 0 if | x | ≥ L ( n ) for some recursive L ( n )) Definition 3.2 { µ m } converges effectively to µ ⇌ { µ m ( f n ) } converges effectively to { µ ( f n ) } for any computable { f n } with recursive compact support e Notation: { µ m } − → µ ~ }

< 13Nancy > 14 weakly: computable sequence f n with compact support → effectively bounded computable { µ ( f n ) } ∃ M ( n ): recursive such that | f n ( x ) | ≤ M ( n ) ew Notation: { µ m } − → µ Proposition 3.3 (1) { µ m } is computable ⇔ { µ m } is weakly computable (2) Assume { µ m } and µ are computable. Then, e ew { µ m } − → µ ⇔ { µ m } − → µ ~

< 13Nancy > 15 4. L´ evy computabilities Definition 4.1 { F n } is said to be L´ evy computable ⇌ {L F n } is computable. Intuitively, L´ evy computability means that we can draw the graph G F effectively. That is, ( ˆ f ( t ) , f ( t )) is a one parametric representation of G F , ( ˆ in the sense of Skotokhod [24]. f ( t ) = t − f ( t )) (Section 6, P. 21) Next Example shows that there exists a probability measure ν : ν is computable G (0) is not a computable real ~ }

< 13Nancy > 16 Example 4.2 (Example 3.4 in [16]) α : one-to-one recursive with non-recursive range. i =1 2 − α ( i ) < 1, d n = � n i =1 2 − α ( i ) ( d 0 = 0) d = � ∞ ∞ n � � 2 − α ( i ) δ 2 − ( i − 1) , ν n = (1 − d n ) δ 0 + 2 − α ( i ) δ 2 − ( i − 1) ν = (1 − d ) δ 0 + i =1 i =1 G and { G n } : the corresponding probability distribution functions 2 ` ¸ (1) 2 ` ¸ (1) 1 ` d 1 2 ` ¸ (2) 2 ` ¸ (2) 1 ` d 2 2 ` ¸ (3) 2 ` ¸ (3) 1 ` d 3 1 ` d 1 ` d 0 1 0 2 Figure 3: Graph of G and L G ~ }

< 13Nancy > 17 The followings holds: • { G n } : monotonically decreasing w.r.t. n , and converges to G • {L G n } : monotonically decreasing w.r.t. n , and converges to L G • L G ( t ) = L G n ( t ) if t ≤ 1 − d or t ≥ 1 − d n +1 + 2 − ( n +1) � = on (1 − d, 1 − d n +1 + 2 − ( n +1) ) • {L G n } is computable We can prove that {L G n } converges to L G effectively uniformly. This implies that L G is computable and G is L´ evy computable. G is not continuous, Fine continuous, not Fine computable, L´ evy computable. ~

< 13Nancy > 18 Theorem 4.3 Assume that { F m } is continuous. Then { F m } is L´ evy computable ⇔ { F m } is computable. R g ( ˆ � � Proposition 4.4 µ ( g ) = R g ( x ) dF ( x ) = f ( t )) d f ( t ) for all g ∈ C κ Theorem 4.5 If { F m } is L´ evy computable, then { µ m } is computable. Lemma 4.6 D a : the probability distribution function of δ a L F is computable, p is positive omputable, a is computable, then (1) L ( pD a ) is computable. (2) L ( pD a + F ) is computable. (3) L F ( t ) ≤ L ( pD a + F )( t ) ≤ p + L F ( t ) . ~

< 13Nancy > 19 R g ( ˆ � � P → C L µ ( g ) = R g ( x ) dF ( x ) = f ( t )) d f ( t ) P → F 1 ❇ ❇ ❇ ❇ w + ❇ ❇ ˜ c; 2 − n w ` ˜ ❇ ❇ c; 2 − n ❇ ❇ ❇ ❇ 0 w + µ ( ˜ w − c,n ) ≤ F ( c ) = µ ( χ ( −∞ .c ] ) ≤ µ ( ˜ c,n ) c c ` 2 ` n c + 2 ` n w + Figure 4: ˜ c,n ( x ) and ˜ w − c,n ( x ) If µ is computable and c is computable, w − w + then µ ( ˜ c,n ) and µ ( ˜ c,n ) are computable. F ( c ) is right (lower) computable but we cannot derive the computability of F ( c ) (+ continuity)