Randomized Computation of Continuous Data: Is Brownian Motion Computable?

TL;DR

This paper investigates the computability of continuous data, specifically Brownian motion, within the framework of Computable Analysis, establishing conditions under which it is computable based on moduli of continuity and extending previous results on randomized computation.

Contribution

It confirms the sufficiency of using the Cantor space of infinite fair coin flips for randomized computation and characterizes the computability of Brownian motion via computable moduli of continuity and their distributions.

Findings

Randomized computation can be confined to Cantor space without loss of generality.

Brownian motion is computable iff its moduli of continuity have a computable distribution.

Extends prior work on biased coin sequences to the case of continuous stochastic processes.

Abstract

We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite FAIR coin flips. This extends [Schr\"oder&Simpson'05] and [Hoyrup&Rojas'09] considering sequences of suitably and adaptively BIASED coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a probability distribution on the space of continuous functions f:[0,1]->R with f(0)=0 whose computability has been conjectured [Davie&Fouch\'e'13; arXiv:1409.4667,S6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure.

Equations

$$\Phi :\mathbb{R}\ni s\mapsto {\int }_{-\infty }^s\exp (-t^2/2)/\sqrt{2\pi }dt\in [0,1]$$

$$F_{>}^{\mu }:(0,1)\ni t\mapsto$$

$$=$$

$$F_{<}^{\mu }:(0,1)\ni t\mapsto$$

$$=$$

$$I_{\vec{0}}\>=\>\big{(}0,\tilde{\gamma}(\vec{0}\circ\mathcal{C})\big{)}\quad\text{as well as}\quad I_{\vec{w}}\>=\>\Big{(}\sum\nolimits_{\vec{v}\boldsymbol{<}\vec{w}}\tilde{\gamma}(\vec{v}\circ\mathcal{C}),\sum\nolimits_{\vec{v}\boldsymbol{\leq}\vec{w}}\tilde{\gamma}(\vec{v}\circ\mathcal{C})\Big{)}$$

$$\mathcal{C}_{\vec{w}}:=\mathcal{C}_{I_{\vec{w}}}\quad\text{ and }\quad F_{n}:\subseteq\mathcal{C}\to\{\textup{{0}},\textup{{1}}\}^{n},\quad F_{n}\big{|}_{\mathcal{C}_{\vec{w}}}:\equiv\vec{w}$$

$$\underset{h\to 0}{\lim }\underset{|s-t|\le h}{\sup }\frac{|W(s)-W(t)|}{\sqrt{2h\ln (1/h)}}=1$$

$$\omega(h,c)\;:=\;\left\{\begin{array}[]{cl}\sqrt{2ch\ln(1/h)}&:\;h\leq 1/Ec\\ y_{c}+(h-1/Ec)\cdot c\cdot\ln(c)/y_{c}&:\;h\geq 1/Ec\end{array}\right.$$

$$\omega(0)=0\quad\wedge\quad\forall x,x^{\prime}\in X:\quad e\big{(}f(x),f(x^{\prime})\big{)}\;\leq\;\omega\big{(}d(x,x^{\prime})\big{)}\enspace.$$

$${\phi }_{n,j}(t)=\{\begin{array}{ll}{2}^{(n-1)/2}\cdot (t-\frac{k-1}{2^n})& \frac{k-1}{2^n}\le t\le \frac{k}{2^n}\\ 2^{(n-1)/2}\cdot (\frac{k+1}{2^n}-t)& \frac{k}{2^n}\le t\le \frac{k+1}{2^n}\\ 0& \text{otherwise}\end{array},0\le k<2j,1\le j\le 2^{n-1}$$

$$W(t{)}^N(\omega )=R_0(\omega )t+\sum _{n=1}^N\sum _{j=1}^{2^{n-1}}{R}_{n,j}(\omega ){\phi }_{n,j}(t)$$

$$W(t{)}^N(\omega )=\sqrt{2}\sum _{i=1}^N{R}_i\frac{\sin (k-\frac{1}{2})\pi t}{(k-\frac{1}{2})\pi }$$

$$W(t{)}^N(\omega )=\frac{S_{\lfloor nt\rfloor }}{\sqrt{N}}$$

$$(C,W)\mapsto \Psi (C,W):=\underset{0\le s,t\le 1}{\max }\omega (|s-t|,C)-|W(s)-W(t)|$$

Full text

Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a probability distribution on the space of continuous functions $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ whose computability has been conjectured [Davie&Fouché’2013; arXiv:1409.4667,§6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure.

1 Introduction

Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult problems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability [DKM*+*94, BMadHS99, CS00, BV04]. Indeed, fair coin flips have been shown computationally universal [Wal77]. Over continuous data, well-known closely connected to topology [Grz57] [Wei00, §2.2+§3], notions of probabilistic computation are more subtle [BGH15, Col15].

1.1 Overview

Section 2 resumes from [SS06] the question of how to represent Borel probability measures. [SS06, Proposition 13] had established that, on ‘reasonable’ spaces, every such distribution can be represented by the distribution of an infinite sequence of coin flips (i.e. over Cantor space) with a suitably and adaptively biased coin. Theorem 2.3 shows that such can in turn be represented by ‘fair’ coins. Lemma 2 characterizes computability of a Borel probability measure on the reals: Necessary and sufficient is that both the lower and upper semi-inverse of its cumulative probability distribution are, respectively, lower and upper semi-computable real functions.

Section 3 approaches the question of whether Brownian Motion (aka Wiener Process), a popular probability distribution on the space of continuous real functions, is computable: in the strong sense of Subsection 2.2 underlying [DF13, MTY13]. Subsection 3.1 recalls several known mathematical characterizations of this distribution, and relates their types of convergence to weaker notions of probabilistic computation [Bos08] while pointing out their differences to the strong sense. It turns out that quantitative continuity of Brownian Motion, captured in terms of some modulus considered as a derived random variable, constitutes the major obstacle: Theorem 3.2 establishes that computability of the probability distribution of any computable such a modulus is both sufficient and necessary for the computability of Brownian Motion. This reduces the conjecture from the probability distribution on a function space to that of an ordinary real random variable.

2 Representing Borel Probability Measures

Recall that a measure space is a triple $(X,\mathcal{A},\mu)$, where $X$ is a non-empty set, $\mathcal{A}$ is a $\sigma$-algebra over $X$, and $\mu$ is a measure on $(X,\mathcal{A})$. For measure spaces $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ and a measurable partial mapping $F:\subseteq X\to Y$, $\nu$ is the pushforward measure of $\mu$ w.r.t. $F$ if \mu\big{(}F^{-1}[V]\big{)} is defined and equal to $\nu(V)$ for every $V\in\mathcal{B}$. In this case we say $F$ realizes $\nu$ on $\mu$ and write $\nu\preccurlyeq\mu$. This notion is similar to, but not in danger of confusion with, [Wei00, Definition 2.3.2]; we will generalize it in Definition 1. Note that realizability is transitive; and a realizer $F$ must have $\operatorname{dom}(F)\in\mathcal{A}$ of measure $\nu(Y)$.

The Type-2 Theory of Effectivity employs Cantor space to encode, and define computation over, any topological T0 space, such as real numbers and continuous real functions [Wei00, §3.2+§4].

Example 2.1

1. a)

Consider the real unit interval $X=[0,1]$ equipped with the $\sigma$-algebra $\mathcal{A}$ of Borel subsets and the Lebesgues probability measure $\lambda$. 2. b)

Consider Cantor space $\mathcal{C}=\{\textup{{0}},\textup{{1}}\}^{\omega}$ equipped with the $\sigma$-algebra $\mathcal{B}$ of Borel subsets and the canonical (=fair coin flip) probability measure $\gamma$: $\gamma(\vec{w}\circ\mathcal{C})=2^{-|\vec{w}|}$, where $|\vec{w}|=n$ denotes the length of $\vec{w}=(w_{0},\ldots,w_{n-1})\in\{\textup{{0}},\textup{{1}}\}^{n}$. 3. c)

The continuous total mapping $\rho_{\textup{b}}:\mathcal{C}\ni\bar{b}\mapsto\sum_{j\geq 0}b_{j}2^{-j-1}\in[0,1]$ realizes $([0,1],\mathcal{A},\lambda)$ on $(\mathcal{C},\mathcal{B},\gamma)$: $([0,1],\mathcal{A},\lambda)\preccurlyeq(\mathcal{C},\mathcal{B},\gamma)$. 4. d)

Consider the real line $\mathbb{R}$ equipped with (the Borel $\sigma$-algebra and) the standard Gaussian/normal probability distribution, realized on $\lambda$ via the partial mapping $G:(0,1)\ni t\mapsto\Phi^{-1}(t)\in\mathbb{R}$ for the cumulative distribution

[TABLE] 5. e)

Consider $[0,1]$ equipped with the Dirac point measue $\delta_{r}$ for some $r\in(0,1)$. It is realized on \big{(}[0,1],\mathcal{A},\lambda\big{)} via the constant function $H:[0,1]\mapsto\{r\}$. 6. f)

*Consider $[0,1]$ equipped with the Cantor measure. It is realized on \big{(}[0,1],\mathcal{A},\lambda\big{)} via the inverse of Devil’s Staircase (aka Cantor–Vitali function). *

Let us combine and generalize the real Items (d), (e), and (f) of Example 2.1:

Lemma 1 (Real Case)

Fix $X=\mathbb{R}$ equipped with some Borel probability measure $\mu$. Recall that its cumulative distribution function \mathbb{R}\ni s\mapsto\mu\big{(}(-\infty,s]\big{)}\in[0,1] is càdlàg (continuous from right with left limits) and non-decreasing, hence upper semi-continuous. On the other hand s\mapsto\mu\big{(}(-\infty,s)\big{)} is càglàd and lower semi-continuous. Now consider the cumulative distribution function’s upper and lower semi-inverse:

[TABLE]

$F^{\mu}_{>}$* is càdlàg and upper-semicontinuous; $F^{\mu}_{<}$ is càglàd and lower-semicontinuous; and both realize $\mu$ on \big{(}[0,1],\mathcal{A},\lambda\big{)}: see Figure 1.*

In the sequel we consider topological spaces, implicitly equipped with the Borel $\sigma$-algebra, and a Borel probability measure. [SS06, Proposition 13] establishes the following:

Fact 2.2

To every 2nd countable T0 space $X$ with Borel probability measure $\mu$ there exists a Borel probability measure $\tilde{\gamma}$ on $\mathcal{C}$ such that $(X,\mu)$ has a continuous partial realizer over $(\mathcal{C},\tilde{\gamma})$.

The metric case is treated in [HR09, Theorem 5.1.1]. We show that the probability measure $\tilde{\gamma}$ on $\mathcal{C}$ can in fact be chosen as the canonical ‘fair’ one:

Theorem 2.3

Every Borel probability measure $\tilde{\gamma}$ on Cantor space $\mathcal{C}$ admits a continuous partial realizer over the ‘fair’ measure $(\mathcal{C},\gamma)$. The realizer is defined on $\mathcal{C}$ with the exception of at most countably many points.

Indeed, Fact 2.2 and transitivity together imply that every 2nd countable T0 space with a Borel probability measure to admit a continuous partial realizer over $(\mathcal{C},\gamma)$. One cannot hope for a total realizer in general, though:

Proposition 1

*Let $\mu$ be a Borel probability measure on $\mathcal{C}$ such that there is some $\bar{v}\in\mathcal{C}$ such that the measure of the basic open set $\bar{v}\,\mathcal{C}$ is *non-dyadic. Then there is no total continuous function $F:\mathcal{C}\to\mathcal{C}$ with $\gamma\circ F^{-1}=\mu$.

2.1 Proofs of Theorem 2.3 and Proposition 1

Proof (Theorem 2.3)

For each open interval $I=(a,b)\subseteq[0,1]$ consider the set $\mathcal{C}_{I}=\rho_{\textup{b}}^{-1}[I]\subseteq\mathcal{C}$ of measure $\gamma(\mathcal{C}_{I})=\lambda(I)$. Note that $\mathcal{C}_{I\cup J}=\mathcal{C}_{I}\cup\mathcal{C}_{J}$ and $\mathcal{C}_{I\cap J}=\mathcal{C}_{I}\cap\mathcal{C}_{J}$. Fix $n\in\mathbb{N}$ and equip $\{\textup{{0}},\textup{{1}}\}^{n}$ with the total lexicographical order; and consider the disjoint open intervals

[TABLE]

of lengths $\lambda(I_{\vec{w}})=\tilde{\gamma}(\vec{w}\circ\mathcal{C})$ for each $\vec{w}\in\{\textup{{0}},\textup{{1}}\}^{n}\setminus\vec{0}$. Since $\tilde{\gamma}$ is a Borel probability measure on $\mathcal{C}$, these lengths add up to $\sum_{\vec{w}}\tilde{\gamma}(\vec{w}\circ\mathcal{C})=1$. Also note that $I_{\vec{w}\,\textup{{0}}},I_{\vec{w}\,\textup{{1}}}\subseteq I_{\vec{w}}$ are disjoint with lengths $\lambda(I_{\vec{w}\,\textup{{0}}})+\lambda(I_{\vec{w}\,\textup{{1}}})=\lambda(I_{\vec{w}})$; and that $I_{\vec{w}}$ may be empty in case $\tilde{\gamma}(\vec{w}\circ\mathcal{C})=0$. Finally abbreviate

[TABLE]

so that $F_{n}$ is defined except for at finitely many arguments (namely the binary encodings of the real interval endpoints) with $F_{n}^{-1}(\vec{w})=\mathcal{C}_{\vec{w}}$ of measure \gamma\big{(}F_{n}^{-1}(\vec{w})\big{)}=\tilde{\gamma}\big{(}\mathcal{C}_{\vec{w}}\big{)}. Since $F_{n+1}(\bar{u})\in F_{n}(\bar{u})\circ\{\textup{{0}},\textup{{1}}\}$, $F(\bar{u}):=\lim_{n}F_{n}(\bar{u})\in\mathcal{C}$ is well-defined (except for at countably many arguments) and continuous with $F^{-1}[\vec{w}\,\mathcal{C}]=\mathcal{C}_{\vec{w}}$ for every $\vec{w}\in\{\textup{{0}},\textup{{1}}\}^{*}$. Hence $\gamma\circ F^{-1}$ coincides with $\tilde{\gamma}$ on the basic clopen subsets of $\mathcal{C}$ and, being Borel measures, also on all Borel subsets. ∎

Proof (Proposition 1)

Suppose that $F$ is such a continuous function. For every $n$ there is a word $\vec{w}_{n}$ of length $n$ such that $F[\vec{w}_{n}\,\mathcal{C}]$ contains an element of $\vec{v}\,\mathcal{C}$ and an element of its complement, because otherwise the preimage $\bar{v}\,\mathcal{C}$ would be the finite union of all open balls $\vec{w}\,\mathcal{C}$ with all $\vec{w}$ of length $n$ satisfying $F[\vec{w}\,\mathcal{C}]\subseteq\vec{v}\,\mathcal{C}$; but the $\gamma$-measure of this union is dyadic. By the fan theorem (or by the fact the $\mathcal{C}$ is a (sequentially) compact space), there is some $\bar{p}\in\mathcal{C}$ and some infinite subset $I$ of $\mathbb{N}$ such that $\vec{w}_{i}$ is a prefix of $\bar{p}$ for all $i\in I$. But $F$ cannot be continuous in the point $\bar{p}$, because no prefix of $\bar{p}$ can tell whether $F(\bar{p})$ is inside or outside the clopen set $\vec{v}\,\mathcal{C}$, a contradiction! ∎

2.2 Computability of Borel Probability Distributions

Of course a realizer in the sense of Theorem 2.3 is usually far from unique. We are interested in those computable with respect to a representation of the space under consideration:

Definition 1

Fix a Borel probability measure $\mu$ on $X$ and a representation $\xi:\subseteq\mathcal{C}\twoheadrightarrow X$ in the sense of TTE [Wei00, §3]. A $\xi$-realizer of $\mu$ is a mapping $G:\subseteq\mathcal{C}\to\operatorname{dom}(\xi)$ such that $\xi\circ G:\subseteq\mathcal{C}\to X$ is a realizer of $\mu$ (over the ‘fair’ measure) in the above sense. Call $\mu$ $\xi$-computable if it has a computable $\xi$-realizer.

Note that $\operatorname{dom}(\xi)\subseteq\mathcal{C}$ is a 2nd countable T0 space, equipped with the pushforward measure of $\xi$: hence Fact 2.2 and Theorem 2.3 together assert that a (possibly uncomputable) $\xi$-realizer exists!

Example 2.4

Recall [Wei00, Definition 2.3.2] computable reduction between representations as well as the importance of admissible ones [Wei00, §3.2], which $\rho_{\textup{b}}$ does not belong to [Wei00, Theorem 4.1.13.6].

1. a)

The identity on $\operatorname{dom}(\rho_{\textup{b}})\subseteq\mathcal{C}$ constitutes a computable $\rho_{\textup{b}}$-realizer of the Lebesgues measure on $[0,1]$ according to Example 2.1c). 2. b)

If $\mu$ is $\xi$-computable and if $\xi\preccurlyeq\xi^{\prime}$ holds, then $\mu$ is also $\xi^{\prime}$-computable. 3. c)

In particular the Lebesgues measure on $[0,1]$ is $\rho$-computable for the admissible representation $\rho$ [Wei00, Theorem 4.1.13.7]. 4. d)

And so is the Gaussian distribution from Example 2.1d) as well as the Cantor distribution from Example 2.1f). The Dirac distribution $\delta_{r}$ is $\rho$-computable  iff  $r$ is $\rho$-computable. 5. e)

For admissible $\xi:\subseteq\mathcal{C}\twoheadrightarrow X$, integration $\mathcal{C}(X)\ni f\mapsto\int_{X}f(x)\,d\mu(x)\in\mathbb{R}$ is \big{(}[\xi\!\to\rho],\rho_{<}\big{)}-computable iff $\mu$ has a computable $\xi$-realizer [Sch07].

Regarding (eq), recall [Wei00, §3.3] that every admissible representation $\xi$ of $X$ induces a canonical admissible representation $[\xi\!\to\rho]$ of the space $\mathcal{C}(X)$ of continuous functions $f:X\to\mathbb{R}$; and recall [Wei00, Lemma 4.1.8] the representation $\rho_{<}$ of $\mathbb{R}$ encoding approximations from below only. It is well-known that every $(\rho,\rho_{<})$-computable function must be lower semi-continuous [WZ00, Zie07]. Resuming Lemma 1, we can now characterize the real case [Wei99]:

Lemma 2

Fix a Borel probability measure $\mu$ on $\mathbb{R}$ with cumulative distribution function and lower and upper semi-inverse $F^{\mu}_{<}$ and $F^{\mu}_{>}$. Then $\mu$ is computable in the sense of Definition 1  iff  both $F^{\mu}_{<}$ is $(\rho,\rho_{<})$-computable and $F^{\mu}_{>}$ is $(\rho,\rho_{>})$-computable.

3 Characterizing Computability of Brownian Motion

1D Brownian Motion aka Wiener Process is a probability measure on the space $X:=\mathcal{C}_{0}[0,1]$ of (i) continuous functions $W:[0,1]\to\mathbb{R}$ satisfying (ii) $W(0)=0$ and characterized by the following properties:

1. iii)

For every $0\leq r<s<t\leq 1$, $W(t)-W(s)$ is independent of $W(r)$. 2. iv)

$W(t)$ is Gaussian normally distributed with mean $W(s)$ and variance $|t-s|$.

Compare [Gal16, KT75, Mal15] for details. Here we approach the question of whether this measure is computable [Fou08, DF13] in the sense of admitting a computable $[\rho\!\to\!\rho]$-realizer; recall Definition 1.

Remark 3.1

The representation $[\rho\!\to\!\rho]$ encodes any $f\in\mathcal{C}[0,1]$ via both (I) its values $f(a/2^{n})$ on the countable dense subset of dyadic rationals $\mathbb{D}:=\bigcup_{n}\mathbb{D}_{n}$, \mathbb{D}_{n}:=\big{\{}0/2^{n},1/2^{n},\ldots,2^{n}/2^{n}\big{\}}, and (II) a binary modulus of continuity of $f$: a sequence $\operatorname{moc}:\mathbb{N}\to\mathbb{N}$ such that $|s-t|\leq 2^{-\operatorname{moc}(n)}$ implies $|W(s)-W(t)|\leq 2^{-n}$; see [Wei00, §6.1].

1. a)

Based on Example 2.1e), Conditions (iii) and (iv) immediately yield an algorithm for computably ‘guessing’ the values W\big{|}_{\mathbb{D}} according to (I) iteratively on-the-fly with respect to the appropriate Gaussian normal distribution in relation to the previous values. However this approach does not allow to then (II) determine $\operatorname{moc}(n)$ within finite time: with small but positive probability, W\big{|}_{\mathbb{D}_{n+m}} may exceed any purported upper bound $\operatorname{moc}(n)$. 2. b)

Conversely first (II) ‘guessing’ $\operatorname{moc}(n)$ requires to know the probability distribution of the random variable $\operatorname{moc}$ exactly: otherwise the resulting Wiener Process will have skewed quantitative continuity. This in turn affects (I) the distribution of W\big{|}_{\mathbb{D}}, with properties (iii) and (iv) now having probabilities conditional to said $\operatorname{moc}$. Recall the following generic (though not necessarily efficient) way of modifying any randomized algorithm to adjust its internal guesses to become conditional to some event $E$: For every independent sample $s$, test shether $g\in E$; if not, discard $s$ and sample again — until obtaining one that complies with $E$. 3. c)

By Lévy’s modulus of continuity theorem, with probability 1 it holds

[TABLE]

The Wiener Process is thus $\alpha$-Hölder continuous for every exponent $\alpha>1/2$, but not for $\alpha=1/2$. 4. d)

More explicitly, abbreviating $E:=\exp(1)$ and $y_{c}:=\sqrt{2\ln(Ec)/c}$, Equation (1) says that, to every $W$ (except for a subset of measure zero) there exists some least $c=c(W)\geq 1$ such that

[TABLE]

depicted in Figure 2 constitutes a parameterized modulus of continuity of $W$ in the following sense: 5. e)

For a function $f:X\to Y$ between metric spaces $(X,d)$ of diameter 1 and $(Y,e)$, a (classical, as opposed to binary) modulus of continuity is a mapping $\omega:[0,1]\to[0,\infty)$ such that it holds

[TABLE]

If $f$ is continuous with compact domain, then it has a modulus of continuity $\omega$. It $X$ is additionally convex, $\omega$ can be chosen subadditive.

The $c\geq 1$ from Item (c) is thus an unbounded real random variable, parameterizing the family of subadditive moduli of continuity from Equation (2) strictly increasing in both arguments.

We can now state our main result characterizing computability of the Wiener Process in terms of computability of the probability distribution of any/all parameterized moduli of continuity:

Theorem 3.2

Let $\omega:[0,1]\times[1,\infty)\to[0,\infty)$ denote any computable (and thus continuous) one-parameter family of subadditive functions strictly increasing in both arguments with $\omega(0,c)\equiv 0$. Suppose that to every Wiener Process $W$ (except for a subset of measure zero) there exists a (necessarily unique) least $c=c(W)\geq 1$ such that $\omega(\>\cdot\>,c)$ constitutes a modulus of continuity of $W$ in the sense of Remark 3.1e). Then the following are equivalent:

• •

The Wiener Process $W$ is computable (formally: has a computable $[\rho\!\to\!\rho]$-realizer).

• •

The random variable $c$ has a computable probability distribution.

• •

There exists a random variable $\tilde{c}$ with computable probability distribution such that $\omega(\>\cdot\>,\tilde{c})$ is a modulus of continuity of $W$ with probability 1.

3.1 Naïve Approaches and their Deficiencies

Before proceeding to the proof of Theorem 3.2, let us report some well-known alternative characterizations of mathematical Brownian Motion and why they do not imply computability.

Example 3.3

For probability spaces $(X,\mu)$ and $(Y,\nu)$, recall that a sequence $R_{n}:\subseteq Y\to X$ of random variables converges almost surely to $R:\subseteq Y\to X$ if the set \big{\{}y:R_{n}(y)\to R(y)\big{\}}\subseteq Y has $\nu$-measure 1.

On the other hand for $(X,d)$ a metric space, uniform almost sure convergence of $R_{n}$ to $R$ means that there exists $U\subseteq\operatorname{dom}(R)\cap\bigcap_{n}\operatorname{dom}(R_{n})$ of $\nu$-measure 1 such that \sup_{y\in U}d\big{(}R_{n}(y),R(y)\big{)}\to 0.

1. a)

Let $\varphi_{0}(t)=t$ and

[TABLE]

denote the Schauder ‘hat’ functions and $R_{n,j}$ independent standard normally distributed random variables. Then following sequence converges to the Wiener Process almost surely:

[TABLE] 2. b)

Let $R_{i}$ be independent standard normally distributed random variables (Example 2.1d). Then following sequence converges to the Wiener Process in mean.

[TABLE] 3. c)

Let $(X_{i})_{i\in\mathbb{N}}$ be independent random variable with mean 0 and variance 1 and $S_{n}=\sum_{i=1}^{n}X_{i}$. Then following sequence converges to the Wiener Process in distribution:

[TABLE]

3.2 Proof of Theorem 3.2

We first record that the hypotheses ensure that $\omega(\>\cdot\>,C)$ does constitute a modulus of continuity for every $C\geq 1$, namely one of $\omega(\>\cdot\>,C)$ itself. Moreover strict monotonicity in $C$ asserts that the measure of all those Wiener Processes $W$ which have $\omega(\>\cdot\>,C)$ as modulus of continuity is strictly increasing and continuous. Hence we can apply Lemma 2 with continuous (as opposed to just càdlàg) cumulative probability distribution and with lower and upper semi-inverse coinciding and continuous.

First suppose the random variable $\tilde{c}$ parameterizing $\omega$ has a computable inverse cumulative probability distribution. Similarly to Example 2.1d), this allows to algorithmically ‘guess’ the value $\tilde{C}$ of $\tilde{c}$ according to said distribution; and computability of $\omega$ can be turned into an (upper bound on the) binary modulus of continuity $\operatorname{moc}$ (II). Regarding (I), having guessed and fixed a modulus of continuity $\omega(\>\cdot\>,\tilde{C})$ affects properties (iii) and (iv) of the Wiener Process. As mentioned in Remark 3.1b), this can be atoned for by discarding guesses for values $W(t)$ that violate $\omega(\>\cdot\>,\tilde{C})$ — but is complicated in our case with undecidable real equality [Wei00, Exercise 4.2.9]. So we make a point of carefully using only strict inequalities, which are at least semi-decidable:

Beginning with $V:=\emptyset$ iteratively/on-demand guess a new value $W(s)$, $s\in\mathbb{D}$, subject to (iii) and (iv). Then check whether it complies with all previously guessed values $W(t)$, $t\in V\subseteq\mathbb{D}$, in satisfying $|W(s)-W(t)|\boldsymbol{<}\omega(|s-t|,\tilde{C})$; and if so, add $s$ to $V$. On the other hand if there is some $t\in V$ with $|W(s)-W(t)|\boldsymbol{>}\omega(|s-t|,\tilde{C})$, then discard and guess again the value of $W(s)$.

Note that the above comparisons exclude and fail in the case $|W(s)-W(t)|\boldsymbol{=}\omega(|s-t|,\tilde{C})$: which occurs only with probability 0, though: The above algorithm thus computes $W$ with probability 1.

Conversely suppose that Brownian Motion has a computable $[\rho\!\to\!\rho]$-realizer $F:\subseteq\mathcal{C}\to\operatorname{dom}([\rho\!\to\!\rho])\subseteq\mathcal{C}$. For each given $C\in\mathbb{R}$ and $\bar{u}\in\operatorname{dom}(F)$ and W:=\big{[}\rho\!\to\!\rho\big{]}(\bar{u}), computability of $\omega$ then implies [Wei00, Corollary 6.2.5] computability of

[TABLE]

and of $W\mapsto\min\{C:\Psi(C,W)\geq 0\}=\max\{C:\Psi(C,W)\leq 0\}=c(W)$ defined for almost all $W$: since $C\mapsto\Psi(C,W)$ is continuous and strictly increasing and unbounded by hypothesis. Again we consider $c(W)=\inf\{C:\Psi(C,W)>0\}=\sup\{C:\Psi(C,W)<0\}$ instead, in order to avoid undecidable real equality. Preimages of open sets can be exhausted (only) from inside [Wei00, Theorem 2.4.5.3], and their measure (only) from below; cmp. Example 2.4e). With $F\circ c$ computable and $F$ a realizer of Brownian Motion, the sought cumulative probability \mathbb{P}\big{[}W:c(W)\leq C\big{]} thus coincides with the fair/canonical measure $\gamma$ of \big{(}F\circ c\big{)}^{-1}\big{[}[0,C)\big{]}\subseteq\mathcal{C}, and with 1-\gamma\Big{(}\big{(}F\circ c\big{)}^{-1}\big{[}(C,\infty)\big{]}\Big{)}: the former yields approximations from below, the latter yield approximations from above, and together they yield approximations up to any given error [Wei00, Lemma 4.1.9]. Note that we do not need $\operatorname{dom}(F)$ to be semi-decidable as it has measure zero anyway. ∎