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Lévy Computability of Probability Distribution Functions Takakazu <13Nancy> 1 L´evyComputability 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 itx # à R e ¹(dx) set of all probability unif. conv. @I measures on R P # à @ 9 @ : " ! @ CF @ @ Bochner @ " ! @ 6 @ @ positive definite @ @ uniformly continuous F (x) = ¹((a; b]) = @ '(0) = 1 @I@ @R @ ¹((¡1; x]) F (b) ¡ F (a) unif. conv. @ # à @ @ ? Inversion @ @ Fine case C Formula @ @ # à L @R @ : 9 " ! F L´evy increasing Lipschitz set of all probability " ! f(`1) = 0; f(+1) = 1 distribution functions on R Discontinuous pointwase conv. at points of continuity Figure 1: The spaces P; F; CL Notations: ¹ $ F $ LF = f (unless oterwise stated) ~ } <13Nancy> 3 0 Introduction 1 Classical Theory 2 L´evyheight and L´evymetric 3 Computability on P 4 L´evycomputabilities 5 L´evyComputability and Fine Computability for Probability Distribution Functions 6 Effective L´evyConvergence ~ } <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 ± 1 is computable, but the corresponding 3 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 ¹ is computable (Example 4.2) We propose L´evycomputability and effective L´evyconvergence 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 ¹(´) = º(´) for 8´ 2 A implies ¹ = º ~ } <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) = 1, F (¡1) = 0 Notations: F: the set of all probability distribution functions C·: the set of all continuous functions with compact support C : the set of all bounded continuous functions b R ¹(f) = R f(x)¹(dx) fg`g: a dense sequence of C· X1 ¡` fg`g-metric : dfg`g(¹; º) = 2 (j¹(g`) ¡ º(g`)j ^ 1) `=1 ~ } <13Nancy> 7 The following converences are equivalent ([1], [6]). (i) ¹m(f) ! ¹(f) 8f 2 C· (vague) (ii) ¹m(f) ! ¹(f) 8f 2 Cb (weak) (iii) lim inf m ¹m(G) ¸ ¹(G) 8G: open domain theory S2007[23] (iv) dfg`g(¹m; ¹) ! 0 fg`g: dense in C· W1999[27][0,1] [0; 1] 0 (iii ) lim inf m ¹m(I) ¸ ¹(I) 8I: open interval W1999[27] Intervals SS2005[22] R (v) 'm(t) ! '(t) for any t (vi) Fm(x) ! F (x) 8x a point of continuity of F , ¹(fxg) = 0, f(¡1; x]g (vii) dL(Fm;F ) ! 0 L´evy (other metrices: cf. [2]) ~ } <13Nancy> 8 2. L´evyheight and L´evymetric L´evy[11], Ito [6] Notations: GF = f(x; y) j F¡(x) · y · F (x)); x 2 Rg (F¡(x) = lim F (y)) y"x 8t 2 R,(x(t); y(t)) = the unique crossing point of X + Y = t with GF Definition 2.1 L´evyheight: LF (t) = y(t) ¯ ¯ Definition 2.2 L´evymetric (distance): dL(F; G) = sup ¯LF (t) ¡LG(t)¯ t2R Definition 2.3 L´evyconvergence: dL(Fn;F ) ! 0 Remark 2.4 dL(F; G) is equal to the L´evy(-Prokhorov) metric, that is, dL(F; G) = inff² > 0 j F (x ¡ ²) ¡ ² · G(x) · F (x + ²) + ², for all xg ~ } <13Nancy> 9 F f(t) = LF (t) a b x F˜`(x) t F˜(x) = t ` f(t) LF Dx ˜ F˜(a) F˜`(b) F˜(b) F˜`(x) t F (x) Figure 2: F and LF Put LF (t) = f(t). Dx = ft j x = t ¡ f(t)g supff(t) j t 2 Dxg ¡ infff(t) j t 2 Dxg = the jump of F at x ~ <13Nancy> 10 Properties of L´evyheight Let CL be the set of functions which satisfy the following: (Li) 0 · f(t) ¡ f(s) · t ¡ s if s · t. (Lii) limt!¡1 f(t) = 0. (Liii) limt!1 f(t) = 1. Proposition 2.5 L: F!CL one-to-one and onto CL : closed convex subset of Cb w.r.t. the sup-norm (distance) d1 Hence, (CL; d1) is complete ~ } <13Nancy> 11 Properties of F˜, F˜¡, LF Dx = [F˜¡(x); F˜(x)] ² F˜(x) = x + F (x) is strictly increasing and continuous LF (t) = F (F˜¡1(t)) ² [F˜¡(x); F˜(x)) \ Range(F ) = Á ² F˜¡1(t) is computable if F is computable ² F (x) = f(sup Dx) ² fˆ(t) = t ¡ f(t) is a nondecreasing continuous function from R onto R <13Nancy> 12 Example 2.6 Dirac measures ±a, Da: prof. dist. function ² Da(x) = 0 if x < a and = 1 if x ¸ a 8 <> 0 if t · a ² LDa(t) = t ¡ a if a · t · 1 + a :> 1 if t ¸ 1 + a ² dL(Da;Db) = d1(LDa; LDb) = ja ¡ bj ^ 1 <13Nancy> 13 3. Computability on P We employ computable notions on R by Pour-El and Richards. Definition 3.1 f¹mg is computable ® f¹m(fn)g is computable for any computable sequence fn with compact support (L(n) s.t. fn(x) = 0 if jxj ¸ L(n) for some recursive L(n)) Definition 3.2 f¹mg converges effectively to ¹ ® f¹m(fn)g converges effectively to f¹(fn)g for any computable ffng with recursive compact support e Notation: f¹mg ¡! ¹ ~ } <13Nancy> 14 weakly: computable sequence fn with compact support ! effectively bounded computable f¹(fn)g 9M(n): recursive such that jfn(x)j · M(n) ew Notation: f¹mg ¡! ¹ Proposition 3.3 (1) f¹mg is computable , f¹mg is weakly computable (2) Assume f¹mg and ¹ are computable. Then, e ew f¹mg ¡! ¹ , f¹mg ¡! ¹ ~ <13Nancy> 15 4. L´evycomputabilities Definition 4.1 fFng is said to be L´evycomputable ® fLFng is computable. Intuitively, L´evycomputability means that we can draw the graph GF effectively. That is, (fˆ(t); f(t)) is a one parametric representation of GF , in the sense of Skotokhod [24]. (fˆ(t) = t ¡ f(t)) (Section 6, P. 21) Next Example shows that there exists a probability measure º: º is computable G(0) is not a computable real ~ } <13Nancy> 16 Example 4.2 (Example 3.4 in [16]) ®: one-to-one recursive with non-recursive range. P1 ¡®(i) Pn ¡®(i) d = i=1 2 < 1, dn = i=1 2 (d0 = 0) X1 Xn ¡®(i) ¡®(i) º = (1 ¡ d)±0 + 2 ±2¡(i¡1) , ºn = (1 ¡ dn)±0 + 2 ±2¡(i¡1) i=1 i=1 G and fGng: the corresponding probability distribution functions `¸(1) 2`¸(1) 2 1 ` d1 2`¸(2) 2`¸(2) 1 ` d2 2`¸(3) 2`¸(3) 1 ` d3 1 ` d 1 ` d 0 1 0 2 Figure 3: Graph of G and LG ~ } <13Nancy> 17 The followings holds: ² fGng: monotonically decreasing w.r.t. n, and converges to G ² fLGng: monotonically decreasing w.r.t. n, and converges to LG ¡(n+1) ² LG(t) = LGn(t) if t · 1 ¡ d or t ¸ 1 ¡ dn+1 + 2 ¡(n+1) 6= on (1 ¡ d; 1 ¡ dn+1 + 2 ) ² fLGng is computable We can prove that fLGng converges to LG effectively uniformly. This implies that LG is computable and G is L´evycomputable. G is not continuous, Fine continuous, not Fine computable, L´evycomputable. ~ <13Nancy> 18 Theorem 4.3 Assume that fFmg is continuous. Then fFmg is L´evycomputable , fFmg is computable. R R ˆ Proposition 4.4 ¹(g) = R g(x)dF (x) = R g(f(t))df(t) for all g 2 C· Theorem 4.5 If fFmg is L´evycomputable, then f¹mg is computable. Lemma 4.6 Da: the probability distribution function of ±a LF is computable, p is positive omputable, a is computable, then (1) L(pDa) is computable. (2) L(pDa + F ) is computable. (3) LF (t) ·L(pDa + F )(t) · p + LF (t). ~ <13Nancy> 19 R R ˆ P!CL ¹(g) = R g(x)dF (x) = R g(f(t))df(t) P!F 1 B B B B w˜+ w˜` B B c;2¡n c;2¡n B B B B 0 ¡ + ¹(w ˜ ) · F (c) = ¹(Â(¡1:c]) · ¹(w ˜ ) c ` 2`n c c + 2`n c;n c;n + ¡ Figure 4: w˜c;n(x) and w˜c;n(x) If ¹ is computable and c is computable, ¡ + then ¹(w ˜c;n) and ¹(w ˜c;n) are computable. F (c) is right (lower) computable but we cannot derive the computability of F (c) (+ continuity) <13Nancy> 20 5. L´evyComputability and Fine Computability for Probability Distribution Functions i i+1 Fine topology is generated by fI(k; i) = [2k ; 2k ) j k 2 N; i 2 Zg J(x; k) is the unique I(k; i) which contains x. Fine computability is defined with respect to this Fine topology. feig is an effective enumaration of dyadic rationals. Definition 5.1 A sequence of functions ffng is said to be Fine computable if it satisfies (i) (Sequential Fine computability) ffn(xm)g is computable for any Fine computable fxmg (ii) (Effective Fine Continuity) There exists a recursive function ®(n; k; i) such that ¡k (ii-a) x 2 J(ei; ®(n; k; i)) ) jfn(x) ¡ fn(ei)j < 2 S1 (ii-b) i=1 J(ei; ®(n; k; i)) = [0; 1) for each n; k ~ } <13Nancy> 21 uniformly Fine computable if ®(n; k; i) does not depend on i Theorem 5.2 fFmg is Fine computable ) fFmg is L´evycomputable First, we prove the following special case.
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