<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 # R e µ(dx) of all probability unif. conv. @I measures on R  P #  @ 9 @ : " !  @ CF @ @ Bochner @ " ! @ 6 @ @ positive definite @ @ uniformly continuous F (x) = µ((a, b]) = @ ’(0) = 1 @I@ @R @ µ((−∞, x]) F (b) − F (a) unif. conv. @ # @ @ ? Inversion @ @ Fine case C Formula @ @ # 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: µ ↔ 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 δ 1 is computable, but the corresponding 3 probability distribution functions is not Fine computable.

We have an example of probability distribution F such that F (0) is not computable, but the corresponding µ 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 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

Cκ: the set of all continuous functions with compact support C : the set of all bounded continuous functions b R µ(f) = R f(x)µ(dx)

{g`}: a dense sequence of Cκ X∞ −` {g`}-metric : d{g`}(µ, ν) = 2 (|µ(g`) − ν(g`)| ∧ 1) `=1 ~ } <13Nancy> 7 The following converences are equivalent ([1], [6]).

(i) µm(f) → µ(f) ∀f ∈ Cκ (vague)

(ii) µm(f) → µ(f) ∀f ∈ Cb (weak)

(iii) lim inf m µm(G) ≥ µ(G) ∀G: open domain theory S2007[23]

(iv) d{g`}(µm, µ) → 0 {g`}: dense in Cκ W1999[27][0,1] [0, 1] 0 (iii ) lim inf m µm(I) ≥ µ(I) ∀I: open interval W1999[27] Intervals SS2005[22] R

(v) ϕm(t) → ϕ(t) for any t

(vi) Fm(x) → F (x) ∀x a point of continuity of F ⇔ µ({x}) = 0, {(−∞, x]}

(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 = {(x, y) | F−(x) ≤ y ≤ F (x)), x ∈ R} (F−(x) = lim F (y)) y↑x

∀t ∈ 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)¯ t∈R

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) = inf{² > 0 | F (x − ²) − ² ≤ G(x) ≤ F (x + ²) + ², for all x}

~ } <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 = {t | x = t − f(t)}

sup{f(t) | t ∈ Dx} − inf{f(t) | t ∈ Dx} = 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→−∞ f(t) = 0.

(Liii) limt→∞ 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) d∞

Hence, (CL, d∞) is complete

~ } <13Nancy> 11

Properties of F˜, F˜−, LF Dx = [F˜−(x), F˜(x)] • F˜(x) = x + F (x) is strictly increasing and continuous LF (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 Dx) • fˆ(t) = t − f(t) is a nondecreasing from R onto R <13Nancy> 12

Example 2.6 Dirac measures δa, Da: prof. dist. function

• Da(x) = 0 if x < a and = 1 if x ≥ a   0 if t ≤ a •LDa(t) = t − a if a ≤ t ≤ 1 + a  1 if t ≥ 1 + a

• dL(Da,Db) = d∞(LDa, LDb) = |a − b| ∧ 1 <13Nancy> 13 3. Computability on P We employ computable notions on R by Pour-El and Richards.

Definition 3.1 {µm} is computable

® {µm(fn)} is computable for any computable sequence fn with compact support

(L(n) s.t. fn(x) = 0 if |x| ≥ L(n) for some recursive L(n))

Definition 3.2 {µm} converges effectively to µ

® {µm(fn)} converges effectively to {µ(fn)}

for any computable {fn} with recursive compact support

e Notation: {µm} −→ µ

~ } <13Nancy> 14 weakly: computable sequence fn with compact support

→ effectively bounded computable {µ(fn)}

∃M(n): recursive such that |fn(x)| ≤ M(n)

ew Notation: {µm} −→ µ

Proposition 3.3

(1) {µm} is computable ⇔ {µm} is weakly computable

(2) Assume {µm} and µ are computable. Then, e ew {µm} −→ µ ⇔ {µm} −→ µ

~ <13Nancy> 15 4. L´evycomputabilities

Definition 4.1 {Fn} is said to be L´evycomputable

® {LFn} 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 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. P∞ −α(i) Pn −α(i) d = i=1 2 < 1, dn = i=1 2 (d0 = 0) X∞ Xn −α(i) −α(i) ν = (1 − d)δ0 + 2 δ2−(i−1) , νn = (1 − dn)δ0 + 2 δ2−(i−1) i=1 i=1 G and {Gn}: 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:

• {Gn}: monotonically decreasing w.r.t. n, and converges to G

• {LGn}: monotonically decreasing w.r.t. n, and converges to LG

−(n+1) • LG(t) = LGn(t) if t ≤ 1 − d or t ≥ 1 − dn+1 + 2 −(n+1) 6= on (1 − d, 1 − dn+1 + 2 )

• {LGn} is computable

We can prove that {LGn} 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 {Fm} is continuous. Then

{Fm} is L´evycomputable ⇔ {Fm} is computable. R R ˆ Proposition 4.4 µ(g) = R g(x)dF (x) = R g(f(t))df(t) for all g ∈ Cκ

Theorem 4.5 If {Fm} is L´evycomputable, then {µm} is computable.

Lemma 4.6 Da: the probability distribution function of δa 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 µ(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 − + µ(w ˜ ) ≤ F (c) = µ(χ(−∞.c]) ≤ µ(w ˜ ) c ` 2`n c c + 2`n c,n c,n

+ − Figure 4: w˜c,n(x) and w˜c,n(x)

If µ is computable and c is computable, − + then µ(w ˜c,n) and µ(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 {I(k, i) = [2k , 2k ) | k ∈ N, i ∈ Z} J(x, k) is the unique I(k, i) which contains x. Fine computability is defined with respect to this Fine topology.

{ei} is an effective enumaration of dyadic rationals.

Definition 5.1 A sequence of functions {fn} is said to be Fine computable if it satisfies (i) (Sequential Fine computability)

{fn(xm)} is computable for any Fine computable {xm} (ii) (Effective Fine Continuity) There exists a recursive function α(n, k, i) such that −k (ii-a) x ∈ J(ei, α(n, k, i)) ⇒ |fn(x) − fn(ei)| < 2 S∞ (ii-b) i=1 J(ei, α(n, k, i)) = [0, 1) for each n, k

~ } <13Nancy> 21 uniformly Fine computable if α(n, k, i) does not depend on i

Theorem 5.2 {Fm} is Fine computable ⇒ {Fm} is L´evycomputable

First, we prove the following special case.

Proposition 5.3 {Fm} is uniformly Fine computable

⇒ {Fm} is L´evycomputable.

Outline of the proof of Proposition 5.3

Lemma 5.4 If {µm} is computable, then there exists a recursive function L(m, k) such that −k c −k µm(wn) > 1 − 2 , equivalently µ(wn) < 2 , for all n ≥ L(m, k). (For a single F ) α(k): mopdulus of effective uniform Fine continuity

note: F−(x) is computable if x is Fine computable

~ <13Nancy> 22

Let {tn} be computable ∀k, the set of fite open intervals (−∞, −L(k) + 2−k), (L(k) − 2−k, ∞), −α(k) −k −α(k) −k (F˜(−L(k) + i2 − 2 , F˜−(−L(k) + (i + 1)2 + 2 · 2 ) (0 ≤ i ≤ 2 · L(k)2α(k)) −α(k) −k −α(k) −k (F˜−(−L(k) + i2 ) − 2 , F˜(−L(k) + i2 ) + 2 ) −α(k) −α(k) −k (F˜(−L(k) + i2 ) − F˜−(−L(k) + i2 ) > 2 ) is an open covering of R

We can define effectively a sequence {rn,k} such that −k |f(tn) − rn,k| < 2 <13Nancy> 23 Outline of the proof of Theorem 5.2

Proposition 5.5 ([15]) Let {fm} be a Fine computable sequences of functions. Define

P n 2 −1 −n χ ϕm,n(x) = j=0 fm(j2 ) I(n,j)(x).

Then, {ϕm,n} Fine converges effectively to {fm}.

We can prove that {Lϕn} converges effectively uniformly to LF . <13Nancy> 24 6. Effective L´evyConvergence

Definition 6.1 {Fm} is said to L´evyconverge effectively to F eL ® dL(Fm,F ) converges effectively to zero {Fm} −→ F

Special case of Skorokhod M1-convergence for GADLAC A pair of functions (λ(t), ξ(t)) is said to be a parametric representation of the graph of GF = {(x, z) | F−(x) ≤ z ≤ F (x), x ∈ R} if GF = {(λ(t), ξ(t) | t ∈ R}, ξ(t) is continuous and λ(t) is continuous and monotonically increasing. ((fˆ(t), f(t)))

{Fm} M1 converges to F : if there exist a parametric representation

(λ(t), ξ(t)) of GF and a sequence of parametric representations

(λm(t), ξm(t)) of {GFm } respectively, such that © ª lim supt |ξm(t) − ξ(t)| + supt |λm(t) − λ(t)| = 0 n→∞

~ } <13Nancy> 25

eL e Theorem 6.2 Let {µm}, µ ∈ P. Then {Fm} −→ F ⇒ {µm} −→ µ R R ˆ use Proposition 4.4 µ(g) = R g(x)dF (x) = R g(f(t))df(t)

Theorem 6.3 Let {µm}, µ ∈ P and {Fm} be L`evycomputable. Then eL {Fm} −→ F ⇒ µ is computable

~ } <13Nancy> 26 Definition 6.4 (Effective d-irrationally pointwise Fine convergence, [15])

Let {Fm}, F be sequentially Fine computable.

{Fm} converges effectively d-irrationally pointwise Fine to F

® {Fm(xn)} converges effectively to {F (xn)} for any Fine computable d-irrational sequence {xn}.

Theorem 6.5 ([16]) Let {Fm}, F be sequentially Fine computable. Assume further that F is effectively Fine continuous. Then, effective convergence of {µm} to µ is equivalent to effective d-irrationally pointwise Fine convergence of {Fm} to F .

Theorem 6.6 Let {Fm}, F be Fine computable. Then, eL {Fm} −→ F ⇔ {Fm} effectively d-irrationally pointwise converges to F

~ } <13Nancy> 27

7. {g`}-metric on P

Proposition 7.1 Let {g`} be a computable sequence in Cκ and

{µm}, {νm} be computable sequences of probability measures. Then,

{d{g`}(µm, νn)} is computable (double) sequence of reals.

Proposition 7.2 Let {g`} be an effective separating set. Then, effective convergence is equivalent to effective {gn}-convergence.

Theorem 7.3 Let S be the set of all computable sequence of probability measures. Then, (P, d{g`}, S) is a metric space with a computability structure in the sense of (cf. Definition [13]).

~ <13Nancy> 28

Example 7.4 {g`}: an example of an effective separating set in Cκ.

(i ` 1)2`m (i + 1)2`m −m i2`m (tm,i = 2 um,i Weihrauch)

Figure 5: Graph of um,i an effective enumeration of the set of all

finite linear combinations with rational coefficients of {um,i}m∈N,i∈Z.

Example 7.5 Example of an effective separating set in (P, d{g`}) an effective enumeration of the set of all

finite convex combinations with rational coefficients of {δi2−m } P2m 22m Proposition 7.6 Define µm = i=0 µ(um,−m+i2−m )δ−m+i2−m . Then, {µm} converges to µ.

Moreover, {µm} is computable and converges effectively to µ, if µ is computable. <13Nancy> 29 Definition 7.7 (Effective compactness, [13]) (X, d, S) is said to be effectively totally bounded if there exist an effective separating set {en} and a recursive function α such that α[(p) −p X = BX(en, 2 ) for all p. n=1 If (X, d, S) is effectively totally bounded and effectively complete, then we say that (X, d, S) is effectively compact.

Proposition 7.8 (P([0, 1]), d{g`}) is effectively compact.

−n Weihrauch [27] had used tn,m = 2 un,m and defined the representation 00 δm. The metric space (P([0, 1]), ρ) is equivalent to (P([0, 1]), dv,{gp}). 0 He also defined representations δm and δm and proved these representations are equivalent. (Theorem 4.2 in [27]). 00 By Kamo [8], d{gp}-computability is equivalent to δm-computability.

~ } <13Nancy> 30 8. Comments on Proofs

Proposition If {µm} is computable and converges effectively to µ, then µ is computable.

The following Lemmas and Proposition are used many times.

Lemma 8.1 (Monotone Lemma, [19])

Let {xn,k} be a computable sequence of reals which converges monotonically to {xn} as k tends to infinity for each n.

Then, {xn} is computable if and only if the convergence is effective.

1 J J wn J 0 JJ `n ` 1 `n n n + 1

Figure 6: Graph of wn <13Nancy> 31 The following Proposition is fundamental.

Proposition 8.2 (Effective tightness of an effectively convergent sequence, Effectivization of Lemma 15.4 in [25])

If a computable {µm} effectively converges to µ, c −k then there exists a recursive function α(k) such that µm(wα(k)) < 2 for all m. C −k It also hold that µm([−α(k) − 1, α(k) + 1] ) < 2 for all m. <13Nancy> 32 References

[1] Billingsley, P., Convergence of Probability Measures. John Wiley & Sons, Inc. 1968. [2] Deza, M. M. and E. Deza, Encyclopedia of Distances (Chap. 14). Springer, 2009. [3] Edgar, G. A., Integral, Probability and Fractal Measures. Springer, 1998. [4] G´acs,Peter. Lecture notes on descriptional complexity and randomness. http://www.cs.bu.edu/faculty/gacs/papers/ait-notes.pdf. [5] Gold, E.M., Limiting recursion, JSL, 30-1 (1965), 28-48 [6] Ito, K. An Introduction to Probability Theory. Cambridge University Press, 1978. (Probability Theory. Iwanami Shotenn, 1991. (in Japanese)) [7] Kamo, H. Effective Dini’s Theorem on Effectively Compact Metric Spaces. ENTCS 120, 73-82, 2005. [8] Kamo, H. Private communication. [9] Kolmogorov, A. N. On Skorokhod Convergence. Teor. Prob. Appl., 1 (1956), 215- 222. [10] Lawson, J. D., Domains, integration and ‘positive analysis’, Math. Struct. in com. Science 14(2004), 815-832. [11] L´evy, P., Th´eoriede L’Addition Variables Al´eatoires.Gautheir-Villars, 1954. <13Nancy> 33 [12] Lu, H. and K. Weihrauch, Computable Riesz Representation for the Dual of C[0; 1], ENTCS 167(2007), 157-177. [13] Mori, T., Y. Tsujii and M. Yasugi. Computability Structures on Metric Spaces. Combinatorics, Complexity and Logic (Proceedings of DMTCS’96), ed. by Bridges et al., 351-362. Springer, 1996. [14] Mori, T., Y. Tsujii and M. Yasugi. Computability of probability distributions and probability distribution functions. Proceedings of the Sixth International Confer- ence on Computability and Complexity in Analysis (DROPS 20), 185-196, 2009. [15] Mori, T., M. Yasugi and Y. Tsujii. Fine convergence of functions and its effec- tivization. Automata, Formal Languages and Algebraic Systems, World Scientific, 2010. [16] Mori, T., Y. Tsujii and M. Yasugi. Fine Computability of Probability Distribu- tion Functions and Computability of Probability Distributions on the Real Line. (manuscript) [17] Mori, T., Y. Tsujii and M. Yasugi. Computability of Probability Distributions and Characteristic Functions. (manuscript) [18] Parthasarathy, K. R. Probability Measures on Metric Spaces. Academic, 1967. [19] Pour-El, M.B. and J. I. Richards. Computability in Analysis and Physics. Springer, 1988. [20] Prokhorov, Yu. V. Convergence of random processes and limit theorems in prob- ability theory. Teor. Prob. Appl., 1 (1956), 157-214. <13Nancy> 34 [21] Rachev, S. T., S. V. Stoyanov and F. J. Fabozzi. A Probability Metrics Approach to Financial Risk Measures. Wiley-Blackwell, 2011. [22] Schr¨oder,M. and A. Simpson. Representing Probability Measures using Proba- bilistic Processes. CCA 2005 (Kyoto), 211-226. [23] Schr¨oder, M., Admissible Representations of Probability Measures. ENTCS 167(2007), 61-78. [24] Skorokhod, A. V. Limit theorems for stochastic processes. Teor. Prob. Appl., 1 (1956), 261-290. [25] Tsurumi, S., Probability Theory. Shibunndou, 1964. (in Japanese) [26] Tsujii, Y., M. Yasugi and T. Mori. Some Properties of the Effective Uniform Topological Space. Computability and Complexity in Analysis, (Lecture Notes in Computer Science 2064), ed. by Blanck, J. et al., 336-356. Springer, 2001. [27] Weihrauch, K. Computability on the probability measures on the Borel sets of the unit interval. TCS 219(1999), 421-437. [28] Weihrauch, K., Computable Analysis. Springer, 2000. [29] Yasugi, M., T. Mori and Y. Tsujii. Effective properties of sets and functions in metric spaces with computability structure. Theoretical Computer Science, 219:467-486, 1999. [30] Yasugi, M., T. Mori and Y. Tsujii. Sequential computability of a function -Effective Fine space and limiting recursion-. JUCS 11-12, 2179-2191, 2005. <13Nancy> 35

9. Characteristic Functions (CCA2012)

Theorem 9.1 If {µm} is computable, then {ϕm} is uniformly computable.

Theorem 9.2 (Effective Glivenko, cf, Theorem 2.6.4 in Ito [6])

Let {ϕm} and ϕ be computable. Then, {µm} converges effectively to µ if {ϕm} converges effectively to ϕ .

Theorem 9.3 (Effectivization of Theorem 2.6.3 in Ito [6]) Let {µm} and µ be computable. Then, {ϕm} converges effectively (compact-uniformly) to ϕ if {µm} converges effectively to µ.

Theorem 9.4 {µm} is computable if {ϕm} is computable.

Theorem 9.4 is the converse to Theorem 9.1. So, we obtain:

Theorem {µm} is computable if and only if {ϕm} is computable. Theorem 9.5 (Effective Bochner’s theorem) In order for ϕ(t) to be a characteristic function of a computable probability measure, it is necessary and sufficient that the following three conditions holds. (i) ϕ is positive definite. (ii) ϕ is computable. (iii) ϕ(0) = 1.

~ } <13Nancy> 36

10. De Moivre-Laplace Central Limit Theorem (CCA2012) The Central Limit Theorem is one of important theorems in probability theory and in statitics.

Let (Ω, B, P, {Xm}) be a realization of Coin Tossing (Bernoulli Trials) with success probability p.

Theorem 10.1 (Effective de Moivre-Laplace) If p is a computable real number, then the sequence of probability measures of random variables m X1 + ··· Xm − mp X X‘ − p Ym = p = √ mpq mp(1 − p) ‘=1 converges effectively to the standard Gaussian probability measure.

p p X `p it q it p Q p‘ p p itYm m it mpq mp ` mq m ψm(t) = E(e ) = ‘=1 E(e ) = (pe + qe )

By Theorems 10.1 and Theorem 9.2, the following hold: Z 2 1 ` t E(f(Ym)) → √ f(t)x2 2 dt 2π R effectively if f is bounded computable. <13Nancy> 37 11. Graphs

¨ ¨¨

¨ ¨¨

¨¨

¨¨

1 1 Figure 7: δ0 + δ 1 2 2 2 <13Nancy> 38

¨ ¨¨

¨ ¨¨

¨ ¨¨ ¨¨

1 1 Figure 8: 2 δ0 + 2 δ1 <13Nancy> 39

¨ ¨¨

¨ ¨¨

¨¨ ¨ ¨ ¨¨

1 1 Figure 9: 2 δ0 + 2 δ2 <13Nancy> 40

(z ; y ) Fm m m (x; F (x)) (x; Fm(x)) Fm (x; Fm(x)) (x; F (x)) (zm; ym)

x F˜m(x) x F˜m(x)

Figure 10: Distance between F (x) and Fm(x) <13Nancy> 41

`k F (xk;‘+1) + 2

`k F (xk;‘+1) ` 2

`k F (xk;‘) + 2

`k F (xk;‘) ` 2

xk,` xk,`+1 F˜(xk,`) F˜(xk,`+1) rk,` rk,`+1 rk,`+2 Ik,` Ik,`+1 F˜−(rk,`+1) F˜(rk,`+1)

Figure 11: L´evyconvergence <13Nancy> 42

h2

h1

h2

Figure 12: Graph of singular G <13Nancy> 43

d d

d 2`¸(2) 2`¸(3) 3 d2 `¸(2) `¸(2) 2 2 LF2 d1 2`¸(1) 2`¸(1) LF1

0 1 0 2

Figure 13: Graph of F and LF <13Nancy> 44

at continuity point of x(t) Ji , Mi - convergence ¡ ¡ ? 6 ¡ ? M1 ¡ pointwise convergence suitable condition on > ZZ~ + - at continuity point of x(t) modulus of (dis)continuity U J1 M2 ZZ~ > J2 ?6 convergence of measure

Figure 14: Relations between Convergences by Skorokhod (case R)

M2 : H-metric between GF and GG

M1 + J2 = J1 <13Nancy> 45 d y(t) [a,c] [b,c] ν[0,1] [y(t)] = 3, ν[0,1][y(t)] = 3 c [a,d] ν[0,1] [y(t)] = 0 b [a,c] [b,c] a ν[0,1] [x(t)] = ν[0,1][x(t)] [a,d] = ν[0,1] [x(t)] = 1 s1 s2 s3 s4 s5 s6 s7s8

d x(t) c

b

s3 s4 s5

Figure 15: Examples of functions in F or D <13Nancy> 46 12. Computability on F and F ↔ P

(classical) convergence of {Fm} to F

Fm(x) → F (x) for ∀x: point of continuity of F

Postulate I (Computability of pdf F ) there is a computable sequence {xn} such that it is dense in R

F is effectively continuous at xn: `¸(n;k) `k ∃α(n, k) s.t. |y − xn| < 2 ⇒ |F (y) − F (xn)| < 2

{F (xn)} is computable

Postulate II (Effective convergene of {Fm}) there is a computable sequence {xn} such that it is dense in R each Fm is effectively continuous at xn

{Fm(xn)} converges effectively uniformly at {xn}

Proposition 12.1 Suppose that there is a computable sequence {xn} such that it is dense in R and {Fm(xn)} is computable. Then {µm} is computable.

Outline of the Proof. for a single F , µ and a single f ∈ C». Take α(k) an effective modulus of uniform continuity of f and an integer L such that f(x) = 0 for |x| ≥ L. <13Nancy> 47

`¸(k+1) ¸(k+1) Let yk;i = −L + i2 (0 ≤ i ≤ 2L2 + 1). We can find effectively n(k, i) such that xn(k;i) ∈ (yk;i`1, yk;i). P ¸(k+1) Then S = 2L2 +1 f(x )(F (x ) − F (x ) converges effectively to R k i=1 n(k;i) n(k;i) n(k;i`1) R f(x)µ(dx). ¯ ¸(k+1) R ¯ P2L2 +1 `k |µ(f) − Sk| = ¯ (f(x) − f(xn(k;i)))µ(dx)¯ < 2 . i=1 (yk;i`1;yk;i]

Proposition 12.2 {µm} is computable and {xn} is computable. If Fm is continuous at {xn}, then Fm(xn) is computable.

`n ` + `n Proof. F (x − 2 ) ≤ µ(w ˜c;n) ≤ F (x) ≤ µ(w ˜c;n) ≤ F (x − 2 ) ` + {µ(w ˜c;n)} and {µ(w ˜c;n)} are computable ` + µ(w ˜c;n) − µ(w ˜c;n) ↓ 0 ⇒ convergence is effective Proposition 12.3 Let LF be computable, x be computable and F be continuous at x. Then F (x) is computable.

Outline of the Proof: On a neighborhood of x, F˜ is strictly increasing and fˆ(t) = t − f(t) is its inverse. (f = LF ) w` , w+ , and Proposition 4.4 may be useful. x;n x;nR R µ(g) = g(x)dF (x) = g(fˆ(t))df(t) for all g ∈ C . R R R » ˆ Lipschitz continuity implies ≤ R g(f)(t)dt if g is nonnegative. <13Nancy> 48

Take g = w` , w+ R x;n x;n R w` (fˆ(t))df(t) = µ(w` ) ≤ F (x) ≤ µ(w+ ) = w+ (fˆ(t))df(t) R x;nR x;nR x;nR R x;n + ˆ ` ˆ + ˆ ` ˆ 0 ≤ R wx;n(f(t))df(t) − R wx;n(f(t))df(t) = R{wx;n(f(t)) − wx;n(f(t))}df(t) R R ˜ `n ≤ {w+ (fˆ(t)) − w` (fˆ(t))}dt = F (x+2 ){w+ (fˆ(t)) − w` (fˆ(t))}dt R x;n x;n F˜(x`2`n) x;n x;n → 0 effectively (?)

Conjecture 12.4 Suppose that there is a computable sequence {xn} such that it is dense in R and {Fm(xn)} is computable. Then {LFm} is computable.

If this conjecture is false, then we can say that L´evycomputability is stronger than computability of the correspondint probability measures. <13Nancy> 49 Other Facts

Fact 12.5 Let {µm} and µ be computable, x be computable, F be continuous at x, and {µm} converge effectively to µ. Then {Fm(x)} converges effectively uniformly to F at x. Or converges effectively continuously at x.

Fact 12.6 Let {Fm} and F be L´evycomputable and x be computable. If {LFm} converges effectively to LF then {Fm(x)} converges effectively uniformly to F at x. R ˆ Need to show R g(f(t))df(t) is computable if g is computable. Put g = un;i. effective convergence ⇔ effective {gn}-convergence ⇒ ??

Fact 12.7 (classical) {Fm} : a sequence of probability distribution functions,

{Fm(x)} : converges (pointwise) to F (x) at every continuity point of F

Then, {Fm(x)} converges uniformly at every continuity point of F

(∵) ∀² > 0 ∃y1 < x < y2 points of continuite of F , |F (y2) − F (y1)| < ²

Fm(y1) → F (y1), Fm(y2) → F (y2),

∃N s.t. m ≥ N, |Fm(y1) − F (y1)| < ², |Fm(y2) − F (y2)| < ²

m ≥ N and |y − x| < min{y2 − x, x − y1}, |Fm(y) − F (x)| < 3² <13Nancy> 50 Other Conjectures

Conjecture 12.8 Let {fm} is a computable sequence in CL and converges monotonically to f ∈ CL. Then the convergence is effective and f is computable.

Effective Dini Theorem 12.9 below by Kamo [7]

Theorem 12.9 Kamo [7]) Let (X, d, S) be an effectively compact metric space. Let

{fn} be a computable sequence of real-valued functions on X and f a computable real-valued function on X. If {fn} converges pointwise monotonically to f as n → ∞, then {fn} converges effectively uniformly to f.

Conjecture 12.10 If X is a effectively compact metric space, then (P(X), dv;fgng) is effectively compact for an effective separating set {gn} of P(X).

Theorem 12.11 (Effective decomposition of unity, Theorem 3 in [29])

Let {On} be an effective local finite r.e. covering of X. Then there exists a computable sequence of functions {fn} which satisfies P1 (i) fn(x) ≥ 0, (ii) fn(x) = 0 if x∈ / On, (iii) n=1 fn(x) = 1. <13Nancy> 51 13. GADLAC

Definition 13.1 A pair of functions (λ(t), ξ(t)) is said to be a parametric representation of the graph of GF = {(x, z) | F`(x) ≤ z ≤ F (x), x ∈ R} if

GF = {(λ(t), ξ(t) | t ∈ R}, ξ(t) is continuous and λ(t) is continuous and monotonically increasing.

Definition 13.2 (Skorokhod) {Fm} SM1 converges to F : if there exist a parametric representation (λ(t), ξ(t)) of GF and a sequence of parametric representations

(λm(t), ξm(t)) of {GFm } respectively, such that © ª lim supt |ξm(t) − ξ(t)| + supt |λm(t) − λ(t)| = 0 n!1

Postulate (1) A gadlac F is said to be SM1- computable if there exist computable functions (λ(t), ξ(t)) which consist a parametric representation of GF . (computable parametric representation)

(2) {Fm} SM1-converges effectively to F : if there exist a computable parametric representation (λ(t), ξ(t)) of GF and a computable sequence of parametric representations (λm(t), ξm(t)) of {GFm } respectively, such that λm(t) and ξm(t)) converges effectively to λ(t) and ξ(t)) respectively. <13Nancy> 52

{Fn} J1-converges to F : There exists a sequence of continuous one-to-one and onto mappings {λ(x)} such that

lim sup |Fn(x) − F (λn(x))| = 0, lim sup |λn(x) − x| = 0. (1) n!1 x n!1 x

This topology is the well known Skorokhod convergence. A equivalent metric which make complete the space of all gadlaces is discussed in Prokhorov [20] Appendix 1 and in Kolmogorov [9] (see also [1], [18]). Since F (λ(x)) is not continuous, it seems difficult to define the corresponding notion of computability.

{Fm} SJ1 converges effectively to F : if there exist computable one-to-one and onto mapping {λm(t)} such that supt |λm(t) − t| and supt |Fm − F (λm(t))| converge effectively to zero.

{Fn} is said to converge uniformly to F at x: For all ² > 0, there exists a δ > 0 such that

lim sup sup |Fn(x) − F (x)| < ². n!1 jy`xj<‹ <13Nancy> 53 Well known properties

(1) If {Fn} converges uniformly to F at any x in some closed finite interval [a, b], then the convergence is uniform on [a, b].

(2) (λ1(t), ξ1(t)) and (λ2(t), ξ2(t)) are parametric representations of some GF , then

there exists a monotonically increasing function u(t) such that λ1(t) = λ2(u(t))

and ξ1(t) = ξ(u(t)). ? not continuous, generalized inverse

(3) If {Fn} converges to F with respect to one of the four convergences, then {Fn} converges uniformly o F at any point of continuity of F .

(4) Let Λ be the set of all continuous one-to-one and onto mappings from [0, 1] to [0, 1]. Then, ¯ © ¯ ª dS(F,G) = inf ² ¯ ∃λ ∈ Λ, sup |t − λ(t)| < ², sup |F (t) − G(λ(t))| < ² © t t ª (=) inf sup |F (t) − G(λ(t))| + sup |t − λ(t)| –2Λ t t

is called the Skorokhod metric. dS-convergence and SJ1-convergence are

equivalent. (D([0, 1]), dS) is not complete. (Skorokhod [24]) <13Nancy> 54 ¯ ¯ ¯ –(t)`–(s) ¯ (5) Let ||λ|| = sups6=t ¯log t`s ¯ and ½ ¯ ¾ ¯ d˜S(F,G) = inf ² ¯ ∃λ, ||λ|| < ², sup |F (t) − G(λ(t))| < ² . t

Then, dS and d˜S are equivalent. (D([0, 1]), d˜S) is complete. (Prokhorov [20], Billingsley [1])   1 1  0 if t < 2 − n n 1 1 1 1 1 1 (6) Let Fn(t) = 2 (t − 2 ) + 2 if 2 − n ≤ t < 2 + n .  1 1 1 if t ≥ 2 + n

Then, {Fn} SM1-converges to D 1 , but the convergence is not SJ2. So, {Fn} does 2 not SJ1-converges to D 1 . 2

Conjecture 13.3 Effective SM1 convergence implies effective uniform convergence at any computable continuity point of F . <13Nancy> 55 14. Two dimensional probability measure Necessary to handle with the Wasserstein Metric

Definition 14.1 The L´evy(L´evy-Prokhorov) metric d˜L(µ, ν) = d˜L(F,G) is defined by

d˜L(F,G) = inf{² > 0 | F (x − ², y − ²) − ² ≤ G(x, y) ≤ F (x + ², y + ²) + ², for all x}

GF = {(x, y, z) | F (x`, y`) ≤ z ≤ F (x, y)}

`s;t = {z + x = s} ∩ {z + y = t},(s, t, v) = GF ∩ `s;t LF (s.t) := v