Yurinskiis Coupling for Martingales Matias D. Cattaneo1Ricardo P. Masini2William G. Underwood3 September 24 2024

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Yurinskii’s Coupling for Martingales

Matias D. Cattaneo1Ricardo P. Masini2William G. Underwood3*

September 24, 2024

Abstract

Yurinskii’s coupling is a popular theoretical tool for non-asymptotic distributional analysis

in mathematical statistics and applied probability, oﬀering a Gaussian strong approximation

with an explicit error bound under easily veriﬁable conditions. Originally stated in

ℓ2

-norm

for sums of independent random vectors, it has recently been extended both to the

ℓp

-norm,

for 1

≤p≤ ∞

, and to vector-valued martingales in

ℓ2

-norm, under some strong conditions.

We present as our main result a Yurinskii coupling for approximate martingales in

ℓp

-norm,

under substantially weaker conditions than those previously imposed. Our formulation further

allows for the coupling variable to follow a more general Gaussian mixture distribution, and

we provide a novel third-order coupling method which gives tighter approximations in certain

settings. We specialize our main result to mixingales, martingales, and independent data, and

derive uniform Gaussian mixture strong approximations for martingale empirical processes.

Applications to nonparametric partitioning-based and local polynomial regression procedures are

provided, alongside central limit theorems for high-dimensional martingale vectors.

Keywords: coupling, strong approximation, mixingales, martingales, dependent data, Gaussian

mixture approximation, time series, empirical processes, uniform inference, series estimation, local

polynomial estimation, central limit theorems.

1Department of Operations Research and Financial Engineering, Princeton University

2Department of Statistics, University of California, Davis

3Statistical Laboratory, University of Cambridge

*Corresponding author: wgu21@cam.ac.uk

arXiv:2210.00362v3 [math.ST] 23 Sep 2024

Contents

1 Introduction 2

1.1 Notation ........................................... 4

2 Main results 5

2.1 User-friendly formulation of the main result ....................... 7

2.2 Mixingales .......................................... 7

2.3 Martingales ......................................... 8

2.4 Independence ........................................ 10

2.5 Stylized example: factor modeling ............................ 11

3 Strong approximation for martingale empirical processes 12

3.1 Motivating example: kernel density estimation ..................... 12

3.2 General result for martingale empirical processes .................... 14

4 Applications to nonparametric regression 17

4.1 Partitioning-based series estimators ............................ 17

4.2 Local polynomial estimators ................................ 19

5 Conclusion 21

A High-dimensional central limit theorems for martingales 26

A.1 Distributional approximation of martingale ℓp-norms .................. 28

B Proofs of main results 30

B.1 Preliminary lemmas .................................... 30

B.2 Main results ......................................... 35

B.3 Strong approximation for martingale empirical processes ................ 42

B.4 Applications to nonparametric regression ......................... 45

1 Introduction

Yurinskii’s coupling (Yurinskii,1978) has proven to be an important theoretical tool for developing

non-asymptotic distributional approximations in mathematical statistics and applied probability.

For a sum

independent zero-mean

-dimensional random vectors, this coupling technique

constructs (on a suitably enlarged probability space) a zero-mean

-dimensional Gaussian vector

which has the same covariance matrix as

and which is close to

in probability, bounding the

discrepancy

∥S−T∥

as a function of

, the choice of norm, and some features of the underlying

distribution. See, for example, Pollard (2002, Chapter 10) for a textbook introduction, and Cs¨org¨o

and R´ev´esz (1981) and Lindvall (1992) for background references.

When compared to other coupling approaches, such as the celebrated Hungarian construction

(Koml´os et al.,1975) or Zaitsev’s coupling (Zaitsev,1987a,b), Yurinskii’s approach stands out for

its simplicity, robustness, and wider applicability, while also oﬀering tighter couplings in some

applications (see below for more discussion and examples). These features have led many scholars

to use Yurinskii’s coupling to study the distributional properties of high-dimensional statistical

procedures in a variety of settings, often with the end goal of developing uncertainty quantiﬁcation

or hypothesis testing methods. For example, in recent years, Yurinskii’s coupling has been used

to construct Gaussian approximations for the suprema of empirical processes (Chernozhukov

et al.,2014b); to establish distribution theory for non-Donsker stochastic

-processes generated in

nonparametric series regression (Belloni et al.,2015); to prove distributional approximations for

high-dimensional

ℓp

-norms (Biau and Mason,2015); to develop distribution theory for vector-valued

martingales (Belloni and Oliveira,2018;Li and Liao,2020); to derive a law of the iterated logarithm

for stochastic gradient descent optimization methods (Anastasiou et al.,2019); to establish uniform

distributional results for nonparametric high-dimensional quantile processes (Belloni et al.,2019); to

develop distribution theory for non-Donsker stochastic t-processes generated in partitioning-based

series regression (Cattaneo et al.,2020); to deduce Bernstein–von Mises theorems in high-dimensional

settings (Ray and van der Vaart,2021); and to develop distribution theory for non-Donsker U-

processes based on dyadic network data (Cattaneo et al.,2024). There are also many other early

applications of Yurinskii’s coupling: Dudley and Philipp (1983) and Dehling (1983) establish

invariance principles for Banach space-valued random variables, and Le Cam (1988) and Sheehy

and Wellner (1992) obtain uniform Donsker results for empirical processes, to name just a few.

This paper presents a new Yurinskii coupling which encompasses and improves upon all of the

results previously available in the literature, oﬀering four new primary features:

(i) It applies to vector-valued approximate martingale data.

(ii) It allows for a Gaussian mixture coupling distribution.

(iii) It imposes no restrictions on degeneracy of the data covariance matrix.

(iv) It establishes a third-order coupling to improve the approximation in certain situations.

Closest to our work are the recent paper by Li and Liao (2020) and the unpublished manuscript

by Belloni and Oliveira (2018), which both investigated distribution theory for martingale data using

Yurinskii’s coupling and related methods. Speciﬁcally, Li and Liao (2020) established a Gaussian

ℓ2

-norm Yurinskii coupling for mixingales and martingales under the assumption that the covariance

structure has a minimum eigenvalue bounded away from zero. As formally demonstrated in this

paper (see Section 3.1), such eigenvalue assumptions can be prohibitively strong in practically

relevant applications. In contrast, our Yurinskii coupling does not impose any restrictions on

covariance degeneracy (iii), in addition to oﬀering several other new features not present in Li and

Liao (2020), including (i),(ii),(iv), and applicability to general

ℓp

-norms. In addition, we correct a

slight technical inaccuracy in their proof relating to the derivation of bounds in probability (see

Remark 2.1).

Belloni and Oliveira (2018) did not establish a Yurinskii coupling for martingales, but rather a

central limit theorem for smooth functions of high-dimensional martingales using the celebrated

second-order Lindeberg method (see Chatterjee,2006, and references therein), explicitly accounting

for covariance degeneracy. As a consequence, their result could be leveraged to deduce a Yurinskii

coupling for martingales with additional, non-trivial technical work (see Appendix Bfor details).

Nevertheless, a Yurinskii coupling derived from Belloni and Oliveira (2018) would not feature (i),(ii),

(iv), or general

ℓp

-norms, as our results do. We discuss further the connections between our work

and the related literature in the upcoming sections, both when introducing our main theoretical

results and when presenting examples and statistical applications.

The most general coupling result of this paper (Theorem 2.1) is presented in Section 2, where

we also specialize it to a slightly weaker yet more user-friendly formulation (Proposition 2.1). Our

Yurinskii coupling for approximate martingales is a strict generalization of all previous Yurinskii cou-

plings available in the literature, oﬀering a Gaussian mixture strong approximation for approximate

martingale vectors in

ℓp

-norm, with an improved rate of approximation when the third moments of

the data are negligible, making no assumptions on the spectrum of the data covariance matrix. A

key technical innovation underlying the proof of Theorem 2.1 is that we explicitly account for the

possibility that the minimum eigenvalue of the variance may be zero, or that its lower bound may

be unknown, with the argument proceeding using a carefully tailored regularization. Establishing a

coupling to a Gaussian mixture distribution is achieved by an appropriate conditioning argument,

leveraging a conditional version of Strassen’s theorem (Chen and Kato,2020, Theorem B.2; Monrad

and Philipp,1991, Theorem 4), along with some related technical work detailed in Appendix B. A

third-order coupling is obtained via a modiﬁcation of a standard smoothing technique for Borel sets

from classical versions of Yurinskii’s coupling (see Lemma B.2 in the appendix), enabling improved

approximation errors whenever third moments are negligible.

In Proposition 2.1, we explicitly tune the parameters of the aforementioned regularization

to obtain a simpler, parameter-free version of Yurinskii’s coupling for approximate martingales,

again oﬀering Gaussian mixture coupling distributions and an improved third-order approximation.

This specialization of our main result takes an agnostic approach to potential singularities in the

data covariance matrix and, as such, may be improved in speciﬁc applications where additional

knowledge of the covariance structure is available. Section 2also presents some further reﬁnements

when additional structure is imposed, deriving Yurinskii couplings for mixingales, martingales, and

independent data as Corollaries 2.1,2.2, and 2.3, respectively. We take the opportunity to discuss

and correct in Remark 2.1 a technical issue which is often neglected (Pollard,2002;Li and Liao,

2020) when using Yurinskii’s coupling to derive bounds in probability. Section 2.5 presents a stylized

example portraying the relevance of our main technical results in the context of canonical factor

models, illustrating the importance of each of our new Yurinskii coupling features (i)–(iv).

Section 3considers a substantive application of our main results: strong approximation of

martingale empirical processes. We begin with the motivating example of canonical kernel density

estimation, demonstrating how Yurinskii’s coupling can be applied, and showing in Lemma 3.1 why

it is essential that we do not place any conditions on the minimum eigenvalue of the variance matrix

(iii). We then present a general-purpose strong approximation for martingale empirical processes

in Proposition 3.1, combining classical results in the empirical process literature (van der Vaart

and Wellner,1996) with our coupling from Corollary 2.2. This statement appears to be the ﬁrst of

its kind for martingale data, and when specialized to independent (and not necessarily identically

distributed) data, it is shown to be superior to the best known comparable strong approximation

result available in the literature (Berthet and Mason,2006). Our improvement comes from using

Yurinskii’s coupling for the

ℓ∞

-norm, where Berthet and Mason (2006) apply Zaitsev’s coupling

(Zaitsev,1987a,b) with the larger ℓ2-norm.

Section 4further illustrates the applicability of our results through two examples in nonparametric

regression estimation. Firstly, we deduce strong approximations for partitioning-based least squares

series estimators with time series data, applying Corollary 2.2 directly and additionally imposing

only a mild mixing condition on the regressors. We show that our Yurinskii coupling for martingale

vectors delivers the same distributional approximation rate as the best known result for independent

data, and discuss how this can be leveraged to yield a feasible statistical inference procedure. We also

show that if the residuals have vanishing conditional third moment, an improved rate of Gaussian

approximation can be established. Secondly, we deduce a strong approximation for local polynomial

estimators with time series data, using our result on martingale empirical processes (Proposition 3.1)

and again imposing a mixing assumption. Appealing to empirical process theory is essential here as,

in contrast with series estimators, local polynomials do not possess certain additive separability

properties. The bandwidth restrictions we require are relatively mild, and, as far as we know, they

have not been improved upon even with independent data.

Section 5concludes the paper. Appendix Ademonstrates how our coupling results can be used to

derive distributional Gaussian approximations (central limit theorems) for possibly high-dimensional

martingale vectors (Proposition A.1). This result complements a recent literature on probability and

statistics studying the same problem but with independent data (see Buzun et al.,2022;Lopes,2022;

Chernozhukov et al.,2023;Kock and Preinerstorfer,2024, and references therein). We also present a

version of this result employing a covariance estimator (Proposition A.2), enabling the construction

of valid high-dimensional conﬁdence sets via a Gaussian multiplier bootstrap. Finally we present

some further results on applications of our theory to deriving distributional approximations for

ℓp-norms of high-dimensional martingale vectors in Appendix A.1.

All proofs are collected in Appendix B, where we also include other technical lemmas of potential

independent interest.

1.1 Notation

We write

∥x∥p

for

p∈

,∞

] to denote the

ℓp

-norm if

is a (possibly random) vector or the induced

operator

ℓp

–

ℓp

-norm if

is a matrix. For

a real-valued random variable and an Orlicz function

, we use

|||X|||ψ

to denote the Orlicz

-norm (van der Vaart and Wellner,1996, Section 2.2) and

|||X|||p

for the

(

) norm where

p∈

,∞

]. For a matrix

, we write

∥M∥max

for the maximum

absolute entry and

∥M∥F

for the Frobenius norm. We denote positive semi-deﬁniteness by

M⪰

and write Idfor the d×didentity matrix.

For scalar sequences

and

, we write

xn≲yn

if there exists a positive constant

such

that

|xn| ≤ C|yn|

for suﬃciently large

. We write

xn≍yn

to indicate both

xn≲yn

and

yn≲xn

Similarly, for random variables

and

, we write

Xn≲PYn

if for every

ε >

0 there exists a

positive constant

such that

(

|Xn| ≥ C|Yn|

)

≤ε

, and write

Xn→PX

for limits in probability.

For real numbers

and

we use

a∨b

max{a, b}

. We write

κ∈Nd

for a multi-index, where

d∈N

{

, . . .}

, and deﬁne

|κ|

j=1 κj

, along with

! =

j=1 κj

!, and

xκ

j=1 xκj

for

x∈Rd.

Since our results concern couplings, some statements must be made on a new or enlarged

probability space. We omit the details of this for clarity of notation, but technicalities are handled

by the Vorob’ev–Berkes–Philipp Theorem (Dudley,1999, Theorem 1.1.10).

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Yurinskii’sCouplingforMartingalesMatiasD.Cattaneo1RicardoP.Masini2WilliamG.Underwood3*September24,2024AbstractYurinskii’scouplingisapopulartheoreticaltoolfornon-asymptoticdistributionalanalysisinmathematicalstatisticsandappliedprobability,offeringaGaussianstrongapproximationwithanexpliciterrorboundu...

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Yurinskiis Coupling for Martingales Matias D. Cattaneo1Ricardo P. Masini2William G. Underwood3 September 24 2024

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