^{1}

^{2}

^{*}

^{1}

^{2}

Edited by: Lixin Shen, Syracuse University, United States

Reviewed by: Gabor J. Szekely, National Science Foundation (NSF), United States; Xin Guo, Hong Kong Polytechnic University, Hong Kong

This article was submitted to Mathematics of Computation and Data Science, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The balance held by Brownian motion between temporal regularity and randomness is embodied in a remarkable way by Levy's forgery of continuous functions. Here we describe how this property can be extended to forge arbitrary dependences between two statistical systems, and then establish a new Brownian independence test based on fluctuating random paths. We also argue that this result allows revisiting the theory of Brownian covariance from a physical perspective and opens the possibility of engineering nonlinear correlation measures from more general functional integrals.

The modern theory of Brownian motion provides an exceptionally successful example of how physical models can have far-reaching consequences beyond their initial field of development. Since its introduction as a model of particle diffusion, Brownian motion has indeed enabled the description of a variety of phenomena in cell biology, neuroscience, engineering, and finance [

Our connection between Brownian motion and independence is motivated by recent developments in statistics, more specifically the unexpected coincidence of two different-looking dependence measures: distance covariance, which characterizes independence fully thanks to its built-in sensitivity to all possible relationships between two random variables [

The

Our strategy to realize this idea consists in establishing that, given any pair

Actually Levy's remarkable theorem, which states that any continuous function can be approximated on a finite interval by generic Brownian paths, provides an obvious starting point of our analysis. Indeed, it stands to reason that if the paths

From a practical standpoint, using Brownian motion to establish independence turns out to be advantageous. Indeed, exploring all bounded continuous transformations exhaustively is realistically impossible. (This practical difficulty motivates the use of reproducing kernel Hilbert spaces, see e.g., Gretton and Györfi [

Here we motivate and describe our main results, with sufficient precision to provide a self-contained presentation of the ideas introduced above while avoiding technical details, which are then developed in the dedicated section 3. We also use here assumptions that are slightly stronger than is necessary, and some generalizations are relegated to the Supplementary Material

Imagine recording the movement of a free Brownian particle in a very large number of trials. In essence, Levy's forgery ensures that one of these traces will follow closely a predefined test trajectory, at least for some time. To formulate this more precisely, let us focus for definiteness on ^{2}〉 = |

Levy's forgery theorem states that this event is generic, i.e., it occurs with probability _{f,δ,T}) > 0 (see Chapter 1, Theorem 38 in [

In all trials though, the particle will eventually drift away to infinity and thus deviate from any bounded test trajectory. Indeed, let us further assume that the function _{f,δ,∞} = ⋂_{T > 0} _{f,δ,T} occurs, the path _{f,δ,∞}) = 0. Hence Levy's forgery theorem does not work on infinite time domains.

To accommodate this asymptotic behavior, we should thus allow the particle to diverge from the test trajectory, at least in a controlled way. Let us recall that the escape to infinity is a.s. slower for Brownian motion than for any movement at constant velocity (which is one way to state the law of large numbers [

whereby the particle is confined to a neighborhood of the test trajectory that

_{f, v, T}) > 0.

An elegant, albeit slightly abstract, proof rests on a short/long time duality between the classes of events (1) and (2), which maps Levy's forgery and this asymptotic version onto each other (see section 3.1). For a more concrete approach, let us focus on the large _{f,v,T} thus merely requires not to outrun deterministic particles moving at speed _{f,v,T}) is close to one for all

We now combine Levy's forgery and the asymptotic version to obtain an extension valid at all timescales. Specifically, let us examine the

In words, the particle is constrained to follow closely the test trajectory for some time but is allowed afterwards to deviate slowly from it (Figure

Extended forgery of continuous functions. This example depicts a test trajectory (smooth curve), its allowed neighborhood (shaded area) and two sample paths, one (solid random walk) illustrating the generic event (3) and the other (dotted random walk), the fact that arbitrary paths have low chances to enter the expanding neighborhoods through the bottlenecks.

_{f,δ,T}) > 0.

This result relies on the suitable integration of a “local” version of the theorem (see section 3.2), but it can also be understood rather intuitively as follows. Imagine for a moment that the events (1) and (2) were independent. Their joint probability would merely be equal to the product of their marginal probabilities, which are positive by Levy's forgery and the asymptotic forgery, and genericity would then follow. Actually they do interact because the associated neighborhoods are connected through the narrow bottlenecks at

The reason lies in the temporal continuity of Brownian motion. A particle staying in the uniform neighborhood while |_{f,δ,T})/_{f,δ,T}) of sample paths _{f,v,T} among all those sample paths _{f,δ,T} should be larger than the unconstrained probability _{f,v,T}), hence the bound (4).

We now turn to the analysis of statistical relations using Brownian motion. Let us fix two random variables

whereby the stochastic covariance cov[

Forgery of statistical dependences. The distribution of the stochastic covariance (black histogram) and its standard deviation [^{4} black dots, Insert

The first step is to ensure that the set (5) is measurable so that its probability is meaningful. Physically, this technical issue is rooted once again in the escape of Brownian particles to infinity. The stochastic covariance can be expressed as a difference of two averages 〈_{2}, i.e., they have finite mean and variance. (See Supplementary Material S2 for a derivation of measurability).

_{2} _{X,Y, f, g, ε}) > 0.

The idea is that one way of realizing the event (5) is to pick sample paths _{f,δ,T},

with _{X,Y, f, g, ε} as well [the necessary condition

To understand Equation (6), imagine first that the random times were bounded with |^{2}〉/^{2}]. The contribution of |

This forgery theorem allows us to probe enough possible relationships to establish statistical dependence or independence, as we explain now. Consider the two probability densities of stochastic covariance shown in Figure

_{2}

To prove sufficiency, we show that the hypothesis cov[_{X,Y, f, g, ε} (Equation 5, shaded area in Figure

Figure

The forgery of statistical dependences provides an alternative approach to the theory of Brownian covariance, hereafter denoted

which is only a slight reformulation of the definition in Székely GJ and Rizzo [_{2} random variables

The first key property is that

The second key aspect is the straightforward sample estimation of _{i},_{i} (

determines an estimator _{2} random variables, this procedure allows to build the estimation theory of Brownian (and thus, distance) covariance from that of the elementary covariance. For instance, the rather intricate algebraic formula for the unbiased sample distance covariance [

because the _{2} convergence hypothesis is sufficient to ensure that averaging over the samples _{i},_{i} commutes with the functional integration over the Brownian paths

Coming back to the implementation of the Brownian independence test, once an estimator

It is noteworthy that our construction of Brownian covariance and its estimator can be generalized by formally replacing the Brownian paths

We now proceed with a more detailed examination of our results. The most technical parts of the proofs are relegated to the Supplementary Material S3.

We sketched in section 2.1 a derivation of the asymptotic forgery theorem using the law of large numbers. A generalization can actually be developed fully (see Supplementary Material S1). However, this particular case enjoys a concise proof based on a symmetry argument.

where the dual _{D} of the function

_{f,v,T}. Explicitly, this event is
_{D}(0) = 0, so (12) coincides with _{fD,v,1/T}. This yields Equation (10).

_{D} is continuous [for _{t∈ℝ} |

We now describe an analytical derivation of the extended forgery theorem that formalizes the intuitive argument given in section 2.1. The ensuing bound for the probability of the joint event (3) does not quite reach that in Equation (4) but is sufficient to ensure genericity.

We are going to focus below on the derivation of

This is sufficient to prove the extended forgery because the backward-time (

and denote their probability by

for all

The first factor in this expectation value denotes the indicator function of the event _{f,δ,T}(

and

They follow from fairly standard arguments about Brownian motion [

The next step is to show that the integrand in the right-hand side of Equation (19) cannot vanish identically, which provides a “local” version of the extended forgery. The full theorem will follow by integration.

for all

By Equations (13) and (14), both lower bounds are positive. The two parts of the theorem are closely related to Levy's forgery and the asymptotic forgery, and we treat them separately.

_{0} − ℓ < _{0} + ℓ with |_{0} − _{0} −

and |_{0}| = |_{0}. The bound (22) follows from these two inclusions by the monotonicity of

_{f, δ, T}. For the second part it is useful to interject here the following simple statements about the function (18):

(i) _{f,δ,T} vanishes identically outside the bottleneck,

(ii) _{f,δ,T} is continuous within the bottleneck (and therefore, it is Borel measurable as well).

The first claim rests on the observation that the set (17) is empty whenever |_{f,δ,T}(

The second claim may appear quite clear as well, since the set (17) should vary continuously with _{n} converging to it. Then the limit event of

The full proof appears rather technical, so we only sketch the key ideas here and relegate the details to the Supplementary Material S3.3. The limit event imposes that sample paths lie within the expanding neighborhood associated with (17) but are allowed to reach its boundary [essentially because the large _{n} yields a nonstrict inequality]. However the latter hitting event a.s. never happens because the typical roughness of Brownian motion forbids meeting a boundary curve without crossing it and thus, leaving the neighborhood (see Supplementary Material S3.4 for this “boundary-crossing law”, which generalizes Lemma 1 on page 283 of [

which is merely the integral formula (19) with the constraint _{f,δ,T}(_{f,δ,T}, so that Equation (23) must hold for at least one point

Indeed, further constraining the paths to _{M}〉 =

It is noteworthy that we stated and proved the extended forgery under the assumption that

We showed in section 2.2 that the forgery of statistical dependences is induced by the extended forgery using the somewhat rough estimate (6) of the covariance error. We provide now an exact upper bound and a complete proof of the theorem.

_{f,δ,T} and

where _{B}(δ,

with _{f}_{t∈ℝ} |_{g}_{t∈ℝ} |

The derivation of this estimate relies on a relatively straightforward application of a series of inequalities and is relegated to the Supplementary Material S3.5. Its significance rests on the fact that, when the polynomial coefficients in (28) are finite, the covariance error can be made arbitrarily small by taking δ,

_{f}, _{g}, 〈|

_{g, δ, T}

_{2} hypothesis and the boundedness assumption on

It is now tempting to invoke the extended forgery theorem, but here the conditions

Note that the _{2} assumption used in section 2.2 was slightly stronger than needed, as made clear by the nested forgery lemma.

To prove that the assertion cov[

Their compatibility, which was indeed central in our derivation, is actually a general property of probability theory and we use it here to provide an alternative, set-theoretic argument.

_{X,Y, f, g, ε} is generic for _{X,Y} occurs a.s. Since the intersection of a generic event and an almost sure event is never empty [if it were empty, the generic event would be a subset of a zero probability event (i.e., the complement of the almost sure event), which is forbidden by monotonicity of

We now exemplify how an explicit estimator of Brownian covariance can be derived from the functional integral (8). Our construction enforces unbiasedness at finite sampling and allows to recover the unbiased sample formula of distance covariance [

_{ij} = |_{i} − _{j}| between the samples _{i} of

where 1 ≤ _{ij} and _{ij}, respectively. With these notations and assuming

This expression differs from the formula given in Székely and Rizzo [

^{2} is then hampered by systematic errors of order 1/

where the primed sum is taken over all distinct indices 1 ≤

This can be proven by averaging Equation (36) over the _{i},_{i}. Indeed, using the identities _{i}_{j}_{i}_{k}〉 = 〈

_{i} = _{i}) and _{i},_{i} are being kept constant. The computation factors into the functional integration of _{i}_{j} = _{i})_{j}) and _{i}_{j} and _{i}_{j} with the autocorrelation functions [

This substitution rule can actually be simplified further to _{i}_{j} → −_{ij}/2 because the terms involving |_{i}| cancel out thanks to the algebraic identity

Similar cancellations also allow to use _{i}_{j} → −_{ij}/2. We thus obtain the unbiased estimator

The equivalence with Equation (35) is not obvious at first sight. To make contact with it, let us fix _{ij}, so Equation (40) can be rewritten as

Finally we recover Equation (35) because _{ij} can be replaced by its U-centering _{ij}. Indeed the extra terms cancel in the sum thanks to the centering property

The author confirms being the sole contributor of this work and approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was carried over in the context of methodological developments for the MEG project at the CUB – Hôpital Erasme (Université libre de Bruxelles, Brussels, Belgium), which is financially supported by the Fonds Erasme (convention Les Voies du Savoir, Fonds Erasme, Brussels, Belgium).

The Supplementary Material for this article can be found online at: