capbtabboxtable[][]

# Bayesian estimation of in-game home team win probability for college
basketball

Jason T. Maddox  Baylor University, Statistical Science, Waco, Texas, U.S.,
e-mail<EMAIL_ADDRESS>Ryan Sides  Texas Woman’s University,
Mathematics and Computer Science, Denton, TX, U.S., e-mail<EMAIL_ADDRESS>Jane L. Harvill  Baylor University, Statistical Science, Waco, Texas, U.S.,
e-mail<EMAIL_ADDRESS>
###### Abstract

Two new Bayesian methods for estimating and predicting in-game home team win
probabilities are proposed. The first method has a prior that adjusts as a
function of lead differential and time elapsed. The second is an adjusted
version of the first, where the adjustment is a linear combination of the
Bayesian estimator with a time-weighted pre-game win probability. The proposed
methods are compared to existing methods, showing the new methods perform
better for both estimation and prediction. The utility is illustrated via an
application to the 2016 NCAA Division 1 Championship game.

Keywords: In-game probability, Pre-game probability, Probability estimation,
Maximum likelihood, Bayesian estimation, Dynamic prior.

##  1 Introduction

Sports analytics is not a new science. Beginning with the work of pioneers
like Bill James, the work spans more than 30 years and a large range of
difficulty. Since the early 2000s, research in statistical methods for sports
analytics has risen dramatically. The review articles of Kubatko et al.,
(2007), Santos-Fernandez et al., (2019), and Terner and Franks, (2021) provide
a fairly comprehensive review for sports analytics for a wide variety of
sports, including basketball.

Generally speaking, models for predicting the outcome of a sporting event can
be classified into two systems: (1) pre-game prediction or (2) in-game, or in-
play, prediction. Pre-game prediction involves determining the outcome of a
game before play begins. In contrast, in-game prediction attempts to use the
progress during a game to determine win probabilities that vary as a function
of in-game variables, for example, elapsed game time, or score difference. The
focus of this paper is in-game prediction, and more specifically, during the
course of the game estimating the probability the home team wins, or the “in-
game win probability.”

Efforts to accurately estimate in-game win probability have long been a
pastime and many different methods for doing so can be found in the
literature. Cooper et al., (1992) collected and analyzed data from 200
basketball games, 100 baseball games, 100 hockey games, and 100 football games
to investigate when, during the course of a game for each of the sports, it is
most likely to know the final outcome of the game. With respect to basketball
they concluded, “late game leaders in basketball go on to win about four in
five times,” and that percentage is “no different in football or hockey.” They
also found that “home teams in basketball were more than three times as likely
as visiting teams to make a fourth-quarter comeback,” and that the home
winning percentage for basketball (64.1%) was the strongest evidence of a
home-team advantage across major American sports.

One of the earlier papers on predicting in-game win probability for baseball
is attributed to Lindsey, (1963) who used the maximum likelihood estimator to
determine the probability the home team wins given the inning the game is in
and the home team’s lead. He looked into how many times a team had won in
prior games when in the same position as the current game was in. That
information was then used to determine that team’s empirical winning
percentage. Up until his work, baseball decisions were based on what would
maximize the scoring output for an inning. Lindsey,’s groundbreaking research
instead focused on determining how each decision would affect the probability
that a team wins instead of only the change in expected score. A more recent
development in estimating in-game win probability is Lock and Nettleton,
(2014) who use random forests that combine pre-play variables to estimate win
probability before each play of an NFL game. Additionally, Ryall, (2011) used
play-by-play data with pre-game Elo rankings to develop a model for Australian
Rules football. The concept of using pre-game power rankings is one that will
be adopted here.

In-game basketball analytics began to surface when Westfall, (1990) developed
a graphical summary of the scoring activity for a basketball game that is a
real-time plot of the score difference versus the elapsed time. The features
of the graph provided easy access to largest leads, lead changes, come-from-
behind activity, and other interesting game features. Over time, models for
forecasting in-game win probability have become more complex. Some are built
on expert predictions, some on betting paradigms, and others on within-game
metrics. Shirley, (2007) modeled a basketball game using a Markov model with
three states and used that model for estimating in-game win probability.
Štrumbelj and Vračar, (2012) improved upon the model of Shirley, by taking
into consideration the strengths of the two teams and estimated transition
probabilities using performance statistics. Vračar et al., (2016) extended the
state description to capture other facets beyond in-game states so that the
transition probabilities become conditional on a broader game context. Bashuk,
(2012) proposed using cumulative win probabilities over the duration of a game
to measure team performance. Using five years of game play, he generated a Win
Probability Index for NCAA basketball. Using that he created an open system to
measure the impact, in terms of win probability added, of each play.

More recently, Chen and Fan, (2018) developed a method for predicting game
outcomes using a functional data approach. Shi and Song, (2019) develop a
discrete-time, finite-state Markov model for the progress of basketball
scores, and use it to conditionally predict the probability the home team wins
or loses by a certain amount. Song and Shi, (2020) present an in-play
prediction model based on the gamma process. They apply a Bayesian dynamic
forecasting procedure that can be used to predict the final score and total
points. Finally, Song et al., (2020) modify the gamma process by employing
betting lines, letting the expectation of the final points total equal the
pre-game betting line.

While the techniques outlined above have various pros and cons, the methods
for estimating the in-game win probability proposed here are aimed at
improving the approaches of Stern, (1994) and Deshpande and Jensen, (2016).
Stern, (1994) modeled the difference between the home and visiting teams’
scores as a Brownian motion process with drift equal to the points in favor of
the home team. This model, which is equivalent to a probit regression model,
results in a relationship between the home team’s lead and the probability of
victory for the home team. His approach is one of the earliest to provide a
mechanism for allowing time to be continuous, and to use that continuity in
modeling the probabilities. Deshpande and Jensen, (2016) extended the work of
Stern, by applying a Bayesian framework to the probit model.

The remainder of the paper is organized as follows. Section 2 contains a more
thorough overview of Stern, (1994) and Deshpande and Jensen, (2016), two
methods for in-game probability designed specifically for NBA games. In
Sections 2.1 and 2.2, a modified enhanced Bayesian approach is proposed that
not only improves in-game predictions compared to existing methods, but also
is suitable for application to NCAA basketball. Section 3 presents a
description of the data used in the study. Following that description is the
result of estimating the in-game home team win probability to over 30,000 NCAA
basketball games, and then using the estimated model to predict the outcome of
over 10,000 NCAA games. To illustrate utility in Section 4, the models are
applied to the 2016 Division 1 NCAA Tournament Championship game between the
University of North Carolina and Villanova University. Finally, Section 5
contains a summary and concluding remarks.

##  2 Estimating In-Game Win Probability

For a specific game, consider the random process that is the home team’s lead
at time $t=0,1,\ldots,2399$, where $t$ is the game time elapsed in seconds. At
a specific time $t$ and for a specific home team lead $\ell$, let $p_{t,\ell}$
denote the in-game probability that the home team will win the game at the end
of regulation. At the beginning of the game, $t=\ell=0$, the estimator of
$p_{0,0}$ is dependent on the method used for estimating in-game win
probability.

When considering multiple games $i=1,2,\ldots,M$, let $Y_{i}=1$ if the home
team wins game $i$ and 0 otherwise. Consider $p_{t,\ell}$ as a continuous
function of $t$ and $\ell$. A classic approach to estimating $p_{t,\ell}$ in
any $(t,\ell)$ cell is the maximum likelihood estimator. Specifically, define
$N_{t,\ell}$ as the number of games in which the home team leads by $\ell$
points after $t$ seconds, and define
$n_{t,\ell}=\sum_{i=1}^{N_{t,\ell}}Y_{i}$, the number of games in the
$(t,\ell)$ cell that the home team wins in regulation. Then on each $(t,\ell)$
cell, the random variable $n_{t,\ell}$ has a binomial($N_{t,\ell},p_{t,\ell})$
distribution. Within each cell, the maximum likelihood estimator of
$p_{t,\ell}$ is $\bar{p}_{t,\ell}=n_{t,\ell}/N_{t,\ell}$. At
$(0,0),~{}\bar{p}_{0,0}=n_{0,0}/N_{0,0}$.

Let $X_{t}$ represent the home team lead after $t$ seconds. Another approach
to estimating $p_{t,\ell}$ is found in Stern, (1994). Stern, estimates in-game
home team win probability via a Brownian motion process with drift $\mu$
points per second home team lead and finite variance $\sigma^{2}$; that is,

$\tilde{p}_{t^{*},\ell}=P(X_{1}>0\,|\,X_{t^{*}}=\ell)=\Phi\left(\frac{\ell+(1-t^{*})\mu}{\sqrt{(1-t^{*})\sigma^{2}}}\right),$
(1)

where $t^{*}\in[0,1)$ represents re-scaled time $t$ and $\Phi(\cdot)$ denotes
the standard normal cumulative distribution function. Although $X_{t}$ is
discrete, the model in equation (1) treats $X_{t}$ as a continuous random
variable. Stern, suggests a continuity correction factor be applied $\ell$,
although he also noted the continuity correction factor results in little
improvement in the model’s performance. He provides empirical evidence that
the Brownian motion model provides a good fit to the score differences when
applied across multiple games. Stern, noted the model (1) can be interpreted
as a probit regression model relating the game outcome to the transformed
variables $\ell/\sqrt{1-t^{*}}$ and $\sqrt{1-t^{*}}$, and with coefficients
$\sigma^{-1}$ and $\mu/\sigma$. At $t^{*}=0$ and
$\ell=0,~{}\tilde{p}_{0,0}=\Phi(\mu/\sigma)$, signifying that $\mu/\sigma$
indicates the magnitude of the home field advantage. Specifying the home team
advantage can be accomplished several ways. Stern, (1994) suggests letting
$\mu$ be the home field advantage in the particular sport, and for basketball,
$\mu=5$ or 6 points. They also note that the home team wins in approximately
55% to 65% of games, and so $\sigma$ can be chosen so that values of
$\mu/\sigma$ are in the 0.12 to 0.39 range.

Some $(t,\ell)$ cells may have small values of $N_{t,\ell}$, which have the
potential to result in estimators of $p_{t,\ell}$ with very large standard
errors. To address this issue, windows can be defined and centered on
$(t,\ell)$ in such a way that the in-game win probability remains relatively
constant across the window. In basketball, since no offensive possession can
result in more than four points the window with respect to $\ell$ can be
reasonably defined as $[\ell-2,\ell+2]$. Moreover, since most offensive
possessions last at least six seconds the width of the time window is taken to
be six. The same notation will be adopted for any
$[t-3,t+3]\times[\ell-2,\ell+2]$ window; that is, $N_{t,\ell}$ is the number
of games in the window in which the home team has led by any value in
$[\ell-2,\ell+2]$ points after any time in $[t-3,t+3]$ seconds and
$n_{t,\ell}=\sum_{i=1}^{N_{t,\ell}}Y_{i}$, distributed as a
binomial($N_{t,\ell},p_{t,\ell})$ random variable.

Bayesian methods provide a third approach to estimating $p_{t,\ell}$. It is
well-known that the beta family is a conjugate prior for estimating
$p_{t,\ell}$. Let $\alpha_{t,\ell}>0$ and $\beta_{t,\ell}>0$ be the shape
parameters of the beta prior on $p_{t,\ell}$. Then for each window within the
$(t,\ell)$ plane, the Bayes estimator of $p_{t,\ell}$ is the mean of the
posterior; specifically

$\hat{p}_{t,\ell}=\frac{n_{t,\ell}+\alpha_{t,\ell}}{N_{t,\ell}+\alpha_{t,\ell}+\beta_{t,\ell}}.$
(2)

For all $t=0,1,\ldots,M$, on all windows centered at $(t,\ell)$, Deshpande and
Jensen, (2016) propose the beta prior

$p_{t,\ell}\sim\left\\{\begin{array}[]{ll}\text{beta}(0,10),&{\text{for
$\ell<-20$,}}\\\ \text{beta}(5,5),&{\text{for $-20\leq\ell\leq 20$,}}\\\
\text{beta}(10,0),&{\text{for $\ell>20$.}}\end{array}\right.$ (3)

The choice of prior depends only on the home team lead $\ell$, and does not
take into account the time remaining in the game $t$.

###  2.1 Dynamic Prior for In-Game Home Team Win Probability

The scale parameters in (3) rely only on the home team lead $\ell$, not taking
into account the interaction between $\ell$ and time remaining. Also notable
is that for $|\ell|>20$, the prior parameters yields an improper prior. In
these cases, the prior may overwhelm the information in the data. To model the
interaction between the home team lead and the time elapsed, we propose
choosing the scale parameters for the beta prior dynamically, as illustrated
in Figure 1 and specified in Table 1.

Table 1: Specification of dynamic beta prior. Color | Prior
---|---
Red | beta(19,1)
Orange | beta(9,1)
Yellow | beta(4,1)
White | beta(1, 1)
Green | beta(1, 4)
Light blue | beta(1, 9)
Blue | beta(1, 19)
Figure 1: Illustration of dynamic beta prior.

The time intervals illustrated in Figure 1 are
$[0,1200),[1200,1800),[1800,2100),[2100,2340)$, and $[2340,2400]$. Observed
home-team differential intervals are time dependent. For the small number of
games when the $|\ell|>50$, the beta(19, 1) or beta(1, 19) prior is applied
accordingly. In the rare instance that a specific $(t,\ell)$ window had no
observed data, for a positive differential, the largest posterior probability
of a home team win is used; for a negative differential, the smallest
posterior probability of a home team win is used. This choice of posterior
estimates is justified since windows with empty cells are those windows with
large score differentials relative to the elapsed time. Moreover, that the
combination hasn’t occurred in over 30,000 games. Therefore, the choice of the
largest or smallest estimate is reasonable. Using the dynamic prior, for any
$(t,\ell)$ in the plane, the dynamic Bayesian estimator of $p_{t,\ell}$ is
given as in equation (2).

The newly proposed choice of prior is proper for all $(t,\ell)$ combinations
and is much less likely to overwhelm the information in the data. More
importantly, the dynamic prior models the in-game home team win probability as
a combination of $(t,\ell)$. As with the prior in (3), the size of the score
differential $|\ell|$ has an affect on the estimated in-game win probability.
However, the newly proposed dynamic prior also models the estimated in-game
win probability as a function of the game time elapsed. In particular, early
in the game (for small $t$), even for a relatively large score differential,
the dynamic prior allows for a larger probability of a comeback than late in
the game, for the same differential.

###  2.2 Adjustment of Bayesian Estimator

For any game, the skill of the two teams playing has a definitive impact on
how likely it is a team will win. If two teams are evenly matched, then
accurately predicting which team will win is more difficult than if one team
is much more skilled than the other. Accordingly, team skill should also be
taken into account when estimating the in-game win probability. None of the
previously described estimators incorporate such a measure. In the following,
we propose an adjusted Bayesian estimator for in-game win probability, where
the adjustment incorporates the skills of the teams in the game.

To determine the adjustment, or the pre-game point spread, the team ratings
and home team advantage are used. The point differential is commonly modeled
using a normal distribution with a standard deviation of 10 points; see for
example, Adams, (2019). The mean for each game is computed using the
difference in the two teams’ ratings with a 3.5 point advantage added for the
home team. Using the normal quantile function, the pre-game point spread can
be converted to a pre-game winning probability, $\hat{p}_{p}$ say, for each
team. For the rare case when a non Division 1 team appeared in the database,
that team is given a rating slightly lower than the lowest ranked Division 1
team. While this may not be accurate for each of these occurrences, no power
rankings on Division 2 and lower teams are available. During the game at each
time $t$, the probability the home team wins $\hat{p}^{*}_{t,\ell}$ is the
weighted average of the pre-game win probability and the current unadjusted
in-game probability, $\hat{p}_{t,\ell}$ found using the dynamic Bayesian model
described in Section 2.1; that is,

$\hat{p}^{*}_{t,\ell}=\left(\frac{2400-t}{2400}\right)\hat{p}_{p}+\left(\frac{t}{2400}\right)\hat{p}_{t,\ell}$
(4)

The adjustment has the following effect. In a tight contest, the in-game win
probability is shifted such that the higher ranked team is predicted to be
more likely to win. When the higher ranked team is winning, the probability
the higher ranked time will win is larger than a lower ranked team that is
winning at the same time by the same margin.

##  3 Comparison of Methods for Estimation and Prediction

In the following, the performances of the MLE, the probit model estimator, the
Bayesian estimator with prior given in (3), and the Bayesian estimator with
dynamic prior in Table 1 and Figure 1 are compared from both the modeling and
prediction perspectives. The games that are used in conducting the study are
from the 2012-2013 seasons up until the 2019-2020 season, which was cut short
due to COVID. In Subsection 3.1, the data is more thoroughly described, and
the data collection process and its challenges are discussed. In Section 3.2
the models are used to compute estimates of in-game home team win probability
using 30,789 games beginning with the 2012-2013 season and ending with the
2017-2018 season. In Section 3.3, the model estimates are used to predict the
outcome of the 10,853 games from the 2018-2019 season and the 2019-2020
season, which was cut short due to COVID. The performances of the models are
compared through Brier’s score and misclassification rates.

###  3.1 NCAA Data, Collection and Challenges

Play-by-play data from ESPN was scraped using _R_ and the package rvest. Since
play-by-play data is not readily available on ESPN’s site for college
basketball games prior to 2012, the data collected begins with the 2012-2013
season and continues through the completion of the 2019-2020 season, when the
season was cut short due to the COVID-19 pandemic.

Due to inconsistencies in data collection and formatting, the data used does
not contain play-by-play data for every game for all teams. From each
conference, a randomly selected team was chosen for inclusion in Table 2,
which shows the percentage of all games played for each season in the data. In
general, smaller schools have a lower percentage of games with available data
than larger schools. However, the lack of information did not appear
systematic, and is not seen as problematic, since the data does include play-
by-play information for many other similar teams.

Table 2: Percentage of games played for which data was collected by ESPN for selected programs from each conference. Team | Conference | 12-13 | 13-14 | 14-15 | 15-16 | 16-17 | 17-18 | 18-19 | 19-20 | Total
---|---|---|---|---|---|---|---|---|---|---
Cent Arkansas | ASUN | 76.7% | 93.1% | 79.3% | 100.0% | 100.0% | 91.4% | 93.9% | 90.3% | 90.6%
UMass Lowell | America East | NA | 89.3% | 89.7% | 96.6% | 93.5% | 93.3% | 96.9% | 90.6% | 92.8%
UCF | American | 90.0% | 93.5% | 93.3% | 96.7% | 97.2% | 96.9% | 87.9% | 96.7% | 94.0%
George Mason | Atlantic 10 | 78.9% | 87.1% | 87.1% | 100.0% | 88.2% | 97.0% | 93.9% | 96.9% | 91.1%
NC State | ACC | 17.1% | 94.4% | 97.2% | 97.0% | 100.0% | 97.0% | 100.0% | 100.0% | 87.8%
Texas Tech | Big 12 | 19.4% | 90.6% | 96.9% | 93.8% | 96.9% | 97.3% | 97.4% | 93.5% | 85.7%
Xavier | Big East | 67.7% | 91.2% | 83.8% | 100.0% | 89.5% | 97.1% | 97.1% | 96.9% | 90.4%
N Arizona | Big Sky | 87.5% | 90.6% | 92.1% | 86.7% | 100.0% | 93.8% | 96.8% | 86.7% | 91.8%
Gardner-Webb | Big South | 85.3% | 90.9% | 94.3% | 87.9% | 93.9% | 96.9% | 85.7% | 90.9% | 90.7%
Ohio State | Big Ten | 10.8% | 85.7% | 85.7% | 100.0% | 100.0% | 97.1% | 97.1% | 100.0% | 84.6%
UC Irvine | Big West | 75.7% | 91.4% | 85.3% | 86.8% | 91.7% | 91.4% | 89.2% | 100.0% | 88.9%
Delaware | Colonial | 60.6% | 94.3% | 90.0% | 86.7% | 100.0% | 100.0% | 90.9% | 90.9% | 89.2%
UNC Charlotte | Conference USA | 75.0% | 90.3% | 90.6% | 93.9% | 96.7% | 93.1% | 93.1% | 100.0% | 91.6%
Oakland | Horizon | 81.8% | 87.9% | 90.9% | 94.3% | 97.1% | 97.0% | 100.0% | 84.8% | 91.7%
Harvard | Ivy | 69.0% | 90.6% | 80.0% | 90.0% | 96.4% | 100.0% | 93.5% | 89.7% | 88.7%
Fairfield | MAAC | 74.3% | 87.5% | 96.8% | 97.0% | 96.8% | 93.9% | 93.5% | 93.8% | 91.7%
Kent State | Mid-American | 80.0% | 78.1% | 91.4% | 93.8% | 91.7% | 97.1% | 100.0% | 96.9% | 91.1%
Morgan State | MEAC | 62.5% | 41.9% | 58.1% | 41.9% | 63.3% | 81.3% | 83.3% | 71.0% | 62.9%
Drake | Missouri Valley | 78.1% | 90.3% | 96.8% | 96.8% | 100.0% | 97.1% | 97.1% | 97.1% | 94.1%
San Diego State | Mountain West | 44.1% | 91.7% | 94.4% | 92.1% | 90.9% | 93.9% | 94.1% | 100.0% | 87.7%
St Francis (PA) | Northeast | 89.7% | 80.6% | 100.0% | 96.7% | 97.1% | 96.8% | 90.9% | 84.4% | 92.0%
SE Missouri St | Ohio Valley | 63.6% | 53.1% | 96.7% | 93.1% | 97.0% | 93.5% | 87.1% | 100.0% | 85.5%
Colorado | Pac-12 | 36.4% | 94.3% | 90.9% | 94.1% | 94.1% | 93.8% | 97.2% | 93.8% | 86.8%
Bucknell | Patriot League | 79.4% | 90.0% | 88.2% | 93.5% | 100.0% | 97.1% | 97.0% | 97.1% | 92.8%
Missouri | SEC | 32.4% | 94.3% | 96.9% | 100.0% | 93.8% | 97.0% | 96.9% | 100.0% | 88.9%
Furman | Southern | 90.3% | 83.3% | 93.9% | 94.3% | 91.4% | 93.9% | 90.9% | 100.0% | 92.3%
McNeese State | Southland | 54.8% | 87.1% | 87.1% | 93.1% | 100.0% | 96.4% | 93.5% | 90.6% | 87.8%
Jackson State | SWAC | 41.4% | 32.3% | 43.8% | 41.7% | 34.4% | 50.0% | 40.6% | 90.6% | 46.8%
Omaha | Summit League | 80.6% | 46.9% | 69.0% | 90.6% | 96.9% | 93.5% | 90.6% | 93.8% | 82.7%
South Alabama | Sun Belt | 83.3% | 90.3% | 97.0% | 87.9% | 96.9% | 96.9% | 97.1% | 96.8% | 93.3%
BYU | West Coast | 58.3% | 94.3% | 94.3% | 97.3% | 94.1% | 100.0% | 90.6% | 100.0% | 91.1%
Chicago State | WAC | 75.8% | 90.6% | 62.5% | 87.5% | 100.0% | 96.9% | 84.4% | 89.7% | 85.9%

Part of the play-by-play data includes the arena where the game was played.
Teams are assigned multiple home arenas in situations where that is
appropriate due to design, i.e., Villanova, or because of a one off, for
example, TCU playing their home games in a different arena for the 2014-2015
season due to renovations. While ESPN is not globally consistent in the way
information is stored, this effect on home-arena determination is rare. It
should also be noted that occasionally teams play in what could be considered
a home game with respect to fan attendance and location. An example of this is
when the University of Kansas plays in Kansas City. Despite these issues, the
database of games is large enough that the overall accuracy of the data used
in the analysis is unaffected.

Finally, to capture each team’s power rating prior to each game, data was
scraped from the website teamrankings.com.111teamrankings.com lacks data for
December 3, 2012. For games played on that date, the team ratings from the
previous day were used. This site was chosen over other systems that may
outperform teamrankings.com because teamrankings.com has historical daily
power ratings available and the other sites do not. An alternative to using
these ratings might be the betting spread from Las Vegas betting lines.

###  3.2 Estimating Home Team In-Game Win Probability

Estimates of in-game home team win probability from the MLE, probit model,
Bayes model using the prior in (3), and the dynamic Bayes model are given in
Figures 2(a) through 2(d). For each $t=0,1,\ldots,2399$ and
$|\ell|=0,1,\ldots,104$ cell, the estimates are illustrated by letting the
color of any $(t,\ell)$ represent the estimated value of $p_{t,\ell}$. The
values $|\ell|=104$ are taken from a game played on Dec. 30, 2013, when
Southern University beat Champion Baptist College by a score of 116-12, the
largest margin of victory for any game included in the data described in
Section 3.1. Historically, the largest margin of defeat in college men’s
basketball occurred on Jan. 12, 1992 when Troy State University, the home
team, beat DeVry University of Atlanta by a score of 258-141. For any cell,
blue represents an estimated in-game home team win probability of
approximately zero, white an estimated probability of approximately 0.5, and
red an estimated probability of approximately one. Examination of these four
figures reveals some common features. In particular, when the score
differential is large, in almost all $(t,\ell)$ cells, all models result in
approximately equal estimated probabilities. Additionally, by the end of the
game the estimated win probability goes to either zero or one for all four
models.

Estimated probabilities using the traditional MLE and probit models are seen
in Figures 2(a) and 2(b), respectively. Both estimates are based on
information in a cell, as opposed to a window, and so are more cells with
missing data than the Bayesian estimates in Figures 2(c) and 2(d).
Additionally, the MLE are less smooth than the other three. In Figure 2(a), up
until $t=1100$, a few windows on the lower bound of the estimates are red,
indicating that even though the home team is down early in the game by a very
large margin, the probability the home teams wins is approximately one. These
cells illustrate one of the problems in using the MLE. The estimates produced
in these cells are based on a single game. In February of 2018, Drexel
University came back from a 34-point deficit to win over the University of
Delaware by a score of 85-83. This comeback is the “largest come-from-behind
win in the history of Division I basketball” (ESPN/Associated Press,, 2018).
Delaware led by a score of 53-19 with 2:36 remaining in the first half. For a
many of the cells associated in the first half of this game, both $N_{t,\ell}$
and $n_{t,\ell}$ are one, resulting in a $\bar{p}_{t,\ell}=1$. For this
specific game, on these specific cells, the MLE performs perfectly on this
training data. However, as evidenced by other games where the home team fell
behind early – just not as badly as Drexel – the home team lost with high
relative frequency. In short, cells with low values of $N_{t,\ell}$ result in
unreliable estimates because the estimators have large standard errors. Rare
events are not as problematic when using the probit model since the drift
$\mu$ and variance $\sigma$ are included in the estimator for each cell. The
Bayesian estimates perform better estimating probabilities for rare events for
two reasons. The first is seen in a type of interpretation of the scale
parameters in the choice of prior. The first scale parameter can be
interpreted as the number of “pseudo-wins” in a cell, and the second scale
parameter as the number of “pseudo-losses.” In this way, the scale parameter
choices can be seen as increasing the number of games in a specific window in
a way that is intuitive for that window. Specifically if the home team is
ahead, then first scale parameter being large effectively acts to increase the
number of wins in that cell. On the other hand, if the home team is behind,
the second scale parameter is large, and thus acting to increase the number of
losses. The second reason rare events do not effect the Bayesian estimates as
drastically as the MLE or as the probit model is due to the Bayesian estimates
being computed over a small window, as opposed to on a cell. Estimates from
the Bayes models, shown in Figures 2(c) and 2(d), are slower to increase the
win probability early in the game compared to the MLE and probit model
estimates. As previously mentioned, because the Bayes estimates are computed
using information in a small window around the cell, their overall appearance
is smoother. The most prominent difference in the two Bayesian estimates is
for score differentials between $-20$ and 20 points. In that range, the Bayes
estimates using the prior in (3) tends to result in moderate probabilities
(0.4 to 0.6) for a early in the game, and for longer period of elapsed time
than the Bayes estimates with the dynamic prior.

Figure 2: Plots illustrating estimated in-game home team win probabilities
based on maximum likelihood (top left), probit model (top right), traditional
Bayes (bottom left), and Bayes with dynamic prior (bottom right).

(a) Maximum likelihood estimates.

(b) Probit model estimates.

(c) Bayesian estimates.

(d) Dynamic Bayesian estimates.

Figures 3(a) through 3(c) illustrate the estimated in-game home team win
probabilities using the adjusted Bayes estimator with dynamic prior. Because
the estimated probability is affected by the pre-game home team win
probability, three values of $\hat{p}_{p}$ were selected to illustrate the
performance of the newly proposed estimator. Of the 30,789 games in the data,
approximately 67% were won by the home team, motivating the choice of
$\hat{p}_{p}=0.67$. The other two choices of pre-game home team win
probability were found by adding and subtracting 0.3 from 0.67. For all of the
30,789 games, the adjusted Bayes estimator of equation (4) was applied. For a
pre-game win probability of $\hat{p}_{p}=0.67$, the adjusted in-game home team
win probabilities are see in Figure 3(a). Figures 3(b) are the estimates for
games with a pre-game win probability of 0.97, and Figure 3(c) for those with
$\hat{p}_{p}=0.37$. A visual comparison of the graphs in Figures 2(a) through
2(c) to Figures 3(a) through 3(c) leads to the conclusion that the unadjusted
estimates perform very differently than the adjusted estimates. In particular,
the adjusted estimates tend to change more slowly than the adjusted estimates.
This is especially true in the first half of the game. The implications of
this slower change is that the adjusted estimates are less likely to produce a
high win probability early in the game for large leads. Later in the game, the
adjusted estimates change more rapidly. An animated Shiny app showing the
evolution of the adjusted dynamic Bayes estimates as a function of
$\hat{p}_{p}$ can be found online at
jasontmaddox.shinyapps.io$\backslash$Win_Probability.

Figure 3: Adjusted dynamic Bayesian estimates for different values of pre-game
win probability.

(a) $\hat{p}_{p}=0.67$.

(b) $\hat{p}_{p}=0.97$.

(c) $\hat{p}_{p}=0.37$.

###  3.3 Assessing the Models Predictive Performance

The five models fitted in Section 3.2 were used to predict the outcome of the
10,853 games in 2018-2019 season and the 2019-2020 season, which was cut short
due to COVID. Two of the more common model evaluation methods used in
classification of binary prediction models are Brier’s score and
misclassification rates. The predictive performances of the MLE, probit model,
Bayes with prior in (3), Bayes with dynamic prior, and adjusted Bayes model
with dynamic prior are compared below using the two measures.

Brier’s score is analogous to the sum of the squared errors in linear
regression and has the advantage of maintaining the continuous information of
all estimations. The square of the difference between the estimated
probability is compared to the observed binary outcome. In the context of in-
game home team win probabilities, this observed binary outcome is denoted
$y_{i}$ and is the observed value of $Y_{i}$ as defined in Section 2. To
interpret Brier’s score, if $Y_{i}=1$ for all $i$, and the predicted
probability is also one for every $i$, then Brier’s score will be zero,
indicating perfect prediction. On the other hand, if for all $i,Y_{i}=0$, and
the estimated probability is one, then Brier’s score will be one, the worst
possible Brier’s score.

To compute Brier’s score in this context, let $\tilde{p}_{t,\ell}$ represent
the estimated in-game home team win probability for any one of the five
methods. For each $(t,\ell)$ cell in the test data set consisting of the
10,853 games, let $N^{*}_{t,\ell}$ represent the number of games in the cell
in which the home team led by $\ell$ points at time $t$; that is,
$N^{*}_{t,\ell}$ is the number of games observed in that cell. Then for non-
missing estimated probabilities, Brier’s score is

$B=\frac{1}{Q}\sum_{t=0}^{2399}\sum_{\ell=-104}^{104}\sum_{j=1}^{N^{*}_{t,l}}\left(\tilde{p}_{t,\ell}-y_{j}\right)^{2},$

where $Q$ is the sum of $N^{*}_{t,\ell}$ in cells without missing
$\tilde{p}_{t,\ell}$. Values of $Q$ for the models are given in Table 3.

Table 3: Number of points in computing Brier’s score and the misclassification rates for five estimation methods. | Model
---|---
| MLE | Probit | Bayes | Dynamic Bayes | Adjusted Dynamic Bayes
$Q$ | 26,034,467 | 26,034,467 | 26,044,933 | 26,047,200 | 26,047,200

Misclassification rates transform the estimated probabilities into binary
indicators. Specifically, a false positive occurs if the estimated in-game
home team win probability is greater than 0.5 and the home team loses
($Y_{i}=0$). A false negative occurs when the estimated in-game home team win
probability is less than 0.5 and the home team wins ($Y_{i}=1$). According to
the same summing operations for calculating Brier’s score, the number of false
positives $FP$ and false negatives $FN$ are counted, and then the two values
are added. Using this, the misclassification rate is defined by

$MR=\frac{FP+FN}{Q}.$

Because Brier’s score maintains the values of the estimated probabilities, it
conveys more information about the models ability to predict than the
misclassification rate. However, due to the sum of squares in Brier’s score,
misclassification rate is more easily interpreted. Both of the evaluation
metrics for each of the three of the four models are provided in Table 4.

Table 4: Evaluation of predictive performances the 2018-2019 and 2019-2020 seasons. | 2018-2019 Season | 2019-2020 Season
---|---|---
Model | $B$ | $MR$ | $B$ | $MR$
MLE | 0.1453 | 0.2183 | 0.1397 | 0.2084
Probit | 0.1452 | 0.2180 | 0.1398 | 0.2086
Bayes | 0.1451 | 0.2182 | 0.1396 | 0.2081
Bayes with dynamic prior | 0.1450 | 0.2182 | 0.1396 | 0.2081
Adjusted dynamic Bayes | 0.1284 | 0.1870 | 0.1261 | 0.1827

With respect to these metrics, the MLE, probit model, Bayes model, and Bayes
model with dynamic prior appear to perform similarly. However, with values of
$Q$ exceeding 26,000,000, these slight differences may be indicative of a true
difference in performance. In contrast, the adjusted Bayesian model with
dynamic prior clearly out performs the other methods, having both lower
Brier’s score, and misclassification rate.

##  4 Application of Model to a Specific Game

Figure 4: Win probability graph for 2016 NCAA Championship Game.

To illustrate utility for a single game, the five estimation techniques were
applied to the 2016 NCAA Division 1 Championship Game between the University
of North Carolina (UNC) and Villanova University. The traces of the estimated
in-game home team win probabilities for all five methods are seen in Figure 4.
The game was played at NRG Stadium in Houston, Texas, a neutral site. UNC was
listed as the home team, as one of the four #1 seeds in the Tournament that
year. Villanova was a #2 seed, but was favored slightly over UNC by
teamrankings.com. The estimated pre-game win probability for UNC to win the
game was $\hat{p}_{p}=0.49$. The first three points of the game were scored
one minute and five seconds into the game by UNC’s Joel Berry II. Nineteen
seconds later, Kris Jenkins of Villanova made a lay-up, and the score was 2-3.
The game stayed close, and at the end of the first half (1,200 seconds into
the game), UNC led Villanova 39-34. An investigation of the five estimators
shows that the MLE, probit model, Bayes, and unadjusted dynamic Bayes all
quickly react to the early scores, bumping up the UNC win probability to as
high as 0.75. At the half, when UNC’s lead was five points, the four
unadjusted methods predict UNC will win with a probability of around 0.85. On
the other hand, the adjusted Bayesian estimate does not react as drastically
as the other four in the early part of the game, because the pre-game win
probability is influencing the returned values.

Five minutes and 52 seconds (1,552 seconds) into the second half, Villanova
had tied the game. At that time, the estimated probability of a UNC win drops
to around 0.5 for all five methods. As the game continued, and the clock wound
down, the five estimates increase and decrease according to the score
differential. In the last few minutes of the game, the change in the five
estimates is more unified in response to the change in score differential than
in the first half of the game. At around time 2,370 seconds elapsed, Villanova
had an eight point lead, and all five models give UNC a very small chance of
winning. UNC pulled within one point with one minute remaining. With six
seconds remaining in the game, Marcus Page of UNC hit a 3-point jump shot to
tie the game at 74-74, and all five estimators jump to a home team win
probability of greater than 0.5. With less than one second remaining, Kris
Jenkins shot a 3-pointer which went through the net after time expired.
Villanova won the championship by a score of 77-74, and all five estimators
drop to zero, showing how Villanova taking the lead with no time remaining
drastically affects the home team’s (UNC) probability of winning.

##  5 Conclusion

For NCAA basketball games, two new methods are proposed for estimating or
predicting in-game home team win probabilities. The first newly proposed
method is a Bayesian estimator with a prior distribution that changes as a
function of lead differential and time elapsed, which was called the Bayesian
estimator with a dynamic prior. The second method adds to the original
estimate a time-weighted adjustment based on pre-game win probability computed
from daily ratings. In this paper, the adjustment was applied only to the
Bayesian estimate with dynamic prior. It is reasonable to conclude the
adjustment would improve the performance of the other estimators, just as it
did the dynamic Bayesian estimator. A comparison of the methods for the
purpose of estimation shows that the two proposed estimates out-performs the
estimates from the three standard methods. For prediction, the adjusted
dynamic Bayesian method out performs the other, based on a comparison of both
Brier’s score and misclassification rates. There are a number of additional
problems to be investigated. First is the determination of an appropriate
method for computing pre-game win probabilities so the same adjustments might
be applied to probability estimates for NBA games. Also of interest is the
effect on the estimators for different window choices. Another consideration
is that the adjustment resulted in a substantial improvement on the prediction
of the adjusted dynamic Bayesian estimator. A different weighting function may
further improve the predictions. Finally, the development of these types of
models for other sports also presents unique challenges that are worth
investigating.

## References

  * Adams, (2019) Adams, T. (2019).  Improving your NCAA Bracket with Statistics.  CRC Press, Boca Raton.
  * Bashuk, (2012) Bashuk, M. (2012).  Using cumulative win probabilities to predict NCAA basketball performance.  In MIT Sloan Sports Analytics Conference.
  * Chen and Fan, (2018) Chen, T. and Fan, Q. (2018).  A functional data approach to model score difference process in professional basketball games.  Journal of Applied Statistics, 45:112–127.
  * Cooper et al., (1992) Cooper, H., DeNeve, K. M., and Mosteller, F. (1992).  Predicting professional game outcomes from intermediate game scores.  CHANCE, 5(3-4):18–22.
  * Deshpande and Jensen, (2016) Deshpande, S. K. and Jensen, S. T. (2016).  Estimating an NBA player’s impact on his team’s chances of winning.  Journal of Quantitative Analysis in Sports, 12(2):51–72.
  * ESPN/Associated Press, (2018) ESPN/Associated Press (2018).  Drexel makes history with 34-point rally to beat delaware.  ESPN.
  * Kubatko et al., (2007) Kubatko, J., Oliver, D., Pelton, K., and Rosenbaum, D. T. (2007).  A starting point for analyzing basketball statistics.  Journal of Quantitative Analysis in Sports, 3(3).
  * Lindsey, (1963) Lindsey, G. R. (1963).  An investigation of strategies in baseball.  Operations Research, 11(4):477–501.
  * Lock and Nettleton, (2014) Lock, D. and Nettleton, D. (2014).  Using random forests to estimate win probability before each play of an NFL game.  Journal of Quantitative Analysis in Sports, 10(2):197–205.
  * Ryall, (2011) Ryall, R. (2011).  Predicting Outcomes in Australian Rules Football.  PhD thesis, Royal Melbourne Institute of Technology University.
  * Santos-Fernandez et al., (2019) Santos-Fernandez, E., Wu, P., and Mengersen, K. L. (2019).  Bayesian statistics meets sports: a comprehensive review.  Journal of Quantitative Analysis in Sports, 15(4):289–312.
  * Shi and Song, (2019) Shi, J. and Song, K. (2019).  A discrete-time and finite-state markov chain based in-play prediction model for NBA basketball matches.  Communications in Statistics: Simulation and Computation.
  * Shirley, (2007) Shirley, K. (2007).  Markov model for basketball.  In Proceedings of the New England Symposium for Statistics in Sports, Boston, MA.
  * Song et al., (2020) Song, K., Gao, Y., and Shi, J. (2020).  Making real-time predictions for NBA basketball games by combining the historical data and bookmaker’s betting line.  Physica A, 547:1–8.
  * Song and Shi, (2020) Song, K. and Shi, J. (2020).  A gamma process based in-play prediction model for National Basketball Association games.  European Journal of Operational Research, 283:706–713.
  * Stern, (1994) Stern, H. S. (1994).  A Brownian motion model for the progress of sports scores.  Journal of the American Statistical Association, 89(427):1128–1134.
  * Štrumbelj and Vračar, (2012) Štrumbelj, E. and Vračar, P. (2012).  Simulating a basketball match with a homogeneous markov model and forecasting the outcome.  International Journal of Forecasting, 28:532–542.
  * Terner and Franks, (2021) Terner, Z. and Franks, A. (2021).  Modeling player and team performance in basketball.  Annual Review of Statistics and Its Applications, 8(1):1–23.
  * Vračar et al., (2016) Vračar, P., Štrumbelj, E., and Kononenko, I. (2016).  Modeling basketball play-by-play data.  Expert Systems with Applications, 44:58–66.
  * Westfall, (1990) Westfall, P. H. (1990).  Graphical presentation of a basketball game.  The American Statistician, 44(4):305–307.


$|\psi_{l}(n)-\psi_{r}(n)|=|(\lambda_{l}(n)-\lambda_{r}(n))+(\delta_{l}(n)-\delta_{r}(n))|.$

Now, $\lim_{|I_{N}|\to\infty}\frac{1}{|I_{N}|}\sum_{n\in
I_{N}}|\lambda_{l}(n)-\lambda_{r}(n)|=0$ and
$\sup_{n\in{\mathbb{Z}}^{L}}|\delta_{r}(n)-\delta_{l}(n)|\leq\frac{1}{l}+\frac{1}{r}$.
Therefore,

(30) $|\psi_{l}(n)-\psi_{r}(n)|\leq\frac{1}{l}+\frac{1}{r}$

for all $n\in{\mathbb{Z}}^{L}$ except potentially a subset
$A\subseteq{\mathbb{Z}}^{L},$ with its characteristic function,
$\mathbbm{1}_{A}(n),$ being a nullsequence. For each $l,r\in{\mathbb{N}}$, the
sequence $\psi_{l}(n)-\psi_{r}(n)$ is a nilsequence, so it follows that
inequality (30) must, in fact, hold for all $n\in{\mathbb{Z}}^{L}$. Hence, the
sequence $(\psi_{l})_{l\in{\mathbb{N}}}$ is a Cauchy sequence in
$\ell^{\infty}({\mathbb{Z}}^{L})$ that consists of $D$-step nilsequences, and
since we already showed that $(\delta_{r})_{r\in{\mathbb{N}}}$ is a Cauchy
sequence in $\ell^{\infty}({\mathbb{Z}}^{L})$ converging to a nullsequence,
from where the conclusion follows. ∎

######  Remark 6.1.

It is worth noting that if the polynomials $p_{1},\dots,p_{k}$ are linear,
then there is an easier proof of Theorem 2.2, where one has $D=k$. The reason
is that, instead of Theorem 2.5, one can use [15, Proposition 1] or [6,
Proposition 6.1] to improve the right hand side of (25) to
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert^{2}_{({\mathbb{Z}}^{d})^{\times(k+1)}}$.

Next, we provide the proof of Theorem 2.4 (the arguments are similar to those
in the proof of [6, Theorem 1.3, Pages 26–27]).

###### Proof of Theorem 2.4.

Using Property (i) and Theorem 2.5, we have that there exists
$D\in{\mathbb{N}}$ such that the left hand side of (8) is 0 if one of
$f_{i}$’s satisfies
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{({\mathbb{Z}}^{d})^{\times D}}=0.$ Thus,
it suffices to show (8) under the assumption that all $f_{i}$’s are measurable
with respect to $Z_{({\mathbb{Z}}^{d})^{\times D}}$. Using Theorem 3.5 and an
approximation argument, we may assume without loss of generality that
$\mathbb{X}$ is a nilsystem. As a consequence of [26, Theorem B] (see also [6,
Theorem 2.9]), Property (ii) is equivalent to (8) on a nilsystem, which
finishes the proof. ∎

Finally, we prove Theorem 2.3. We start with proving that (C1) implies (C2).
In fact, we show the following more general result:

######  Proposition 6.2.

Let $d,k,L\in\mathbb{N},$
$q_{1},\dots,q_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be polynomials, and
$\mathbb{X}=(X,\mathcal{B},\mu,$ $(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\mathbb{Z}^{d}$-system. Suppose that
$(T_{q_{1}(n)},\dots,T_{q_{k}(n)})_{n\in{\mathbb{Z}}^{L}}$ is jointly ergodic
for $\mu$. Then for all $1\leq i,j\leq k,i\neq j$, we have that
$(T_{q_{i}(n)-q_{j}(n)})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu$.

Furthermore, if there exist polynomials
$p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}$ and
$v_{1},\dots,v_{k}\in{\mathbb{Z}}^{d}$ such that $q_{i}(n)=p_{i}(n)v_{i}$ for
all $1\leq i\leq k$, then $(T_{p_{1}(n)v_{1}}\times\dots\times
T_{p_{k}(n)v_{k}})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu^{\otimes k}$.

###### Proof.

The sequence $(T_{q_{i}(n)-q_{j}(n)})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for
$\mu$ for all $1\leq i\neq j\leq k$ by an argument similar to the one given in
the proof of [6, Proposition 5.3], so, we choose to omit the details.

We now assume that $q_{i}(n)=p_{i}(n)v_{i}$ for all $1\leq i\leq k$ and show
that $(T_{p_{1}(n)v_{1}}\times\dots\times
T_{p_{k}(n)v_{k}})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu^{\otimes k}$.
It suffices to show that for all $f_{i}\in L^{\infty}(\mu),1\leq i\leq k$ with
$\prod_{i=1}^{k}\int_{X}f_{i}\,d\mu=0$, we have that

(31) $\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\bigg{\|}\mathbb{E}_{n\in
I_{N}}\bigotimes_{i=1}^{k}T_{p_{i}(n)v_{i}}f_{i}\bigg{\|}_{L^{2}(\mu^{\otimes
k})}=0.$

Claim. If
$\mathbb{E}(f_{i}|Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}(\mathbb{X}))=0$ for
some $1\leq i\leq k$, then (31) holds.

We may assume without loss of generality that
$\deg(p_{1})\geq\deg(p_{2})\geq\dots\geq\deg(p_{k})$. Suppose that we have
shown that for some $1\leq k_{0}\leq k$, (31) holds if
$\mathbb{E}(f_{i}|Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}(\mathbb{X}))=0$ for
some $1\leq i\leq k_{0}-1$, where the case $k_{0}=1$ is understood to be
always true. It suffices to show that (31) holds if
$\mathbb{E}(f_{k_{0}}|Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}(\mathbb{X}))=0$.

By the induction hypothesis, we may assume without loss of generality that
$f_{i}$ is $Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}(\mathbb{X})$-measurable for
all $1\leq i\leq k_{0}-1$. By [6, Lemma 2.7], we can approximate each $f_{i}$
in $L^{2}(\mu)$ by an eigenfunction of $\mathbb{X}$. By multi-linearity, we
may assume without loss of generality that each $f_{i},1\leq i\leq k_{0}-1$ is
a non-constant eigenfunction of $\mathbb{X}$ given by
$T_{n}f_{i}=\exp(\lambda_{i}(n))f_{i}$ for all $n\in\mathbb{Z}^{d}$ and some
group homomorphism $\lambda_{i}\colon\mathbb{Z}^{d}\to{\mathbb{R}}$, and that
$f_{i}(x)\neq 0$ $\mu$-a.e $x\in X$. Then the left hand side of (31) is equal
to

(32) $\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\bigg{\|}\mathbb{E}_{n\in
I_{N}}\exp(P(n))\bigotimes_{i=1}^{k_{0}-1}f_{i}\bigotimes_{i=k_{0}}^{k}T_{p_{i}(n)v_{i}}f_{i}\bigg{\|}_{L^{2}(\mu^{\otimes
k})},$

where $P(n)\coloneqq\sum_{i=1}^{k_{0}-1}\lambda_{i}(v_{i})p_{i}(n).$ Denote
$P(n)=\sum_{j=0}^{\deg(p_{1})}Q_{j}(n)$, where $Q_{j}$ is a homogeneous
polynomial of degree $j$ for all $0\leq j\leq\deg(p_{1})$. Then

(33)
$\begin{split}\Delta^{K}P(n,h_{1},\dots,h_{K})=\sum_{j=K}^{\deg(p_{1})}\Delta^{K}Q_{j}(n,h_{1},\dots,h_{K}).\end{split}$

We first consider the case when $Q_{j}(n)\notin\mathbb{Q}[n]+{\mathbb{R}}$ for
some $\deg(p_{k_{0}})+1\leq j\leq\deg(p_{1})$. In this case, let
$K=\deg(p_{k_{0}})$ in (33). Since $\Delta^{K}p_{i}(n,h_{1},\dots,h_{K})$ is
constant in $n$ for all $k_{0}\leq i\leq k$, by Lemma 4.2, to show that (32)
is 0, it suffices to show that

(34)
$\begin{split}&\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{K}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}|\mathbb{E}_{n\in
I_{N}}\exp(\Delta^{K}P(n,h_{1},\dots,h_{K}))|=0\end{split}$

(see Definition 3.13 for the definition of the polynomial $\Delta^{K}P$).

As $Q_{j}(n)\notin\mathbb{Q}[n]+{\mathbb{R}}$ for some $\deg(p_{k_{0}})+1\leq
j\leq\deg(p_{1})$, Lemma 3.14 implies that
$\Delta^{K}Q_{j}(\cdot,h_{1},\dots,h_{K})\notin\mathbb{Q}[n]+{\mathbb{R}}$ for
a set of $(h_{1},\dots,h_{K})$ of density 1. By Weyl’s criterion and (33), we
have that (34) holds and thus (31) holds.

We now consider the case when $Q_{j}(n)\in\mathbb{Q}[n]+{\mathbb{R}}$ for all
$K+1\leq j\leq\deg(p_{1})$. Let $P^{\prime}(n)=\sum_{j=0}^{K}Q_{j}(n)$. It is
not hard to see that there exists $Q\in{\mathbb{N}}$ such that for all
$r\in\\{0,\dots,Q-1\\}^{L}$ and $n\in\mathbb{Z}^{L}$, we have that

$P(Qn+r)-P^{\prime}(Qn+r)=P(r)-P^{\prime}(r).$

By (32), to show (31), it suffices to show that for all
$r\in\\{0,\dots,Q-1\\}^{L}$, we have that

(35) $\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Biggl{\|}\mathbb{E}_{n\in
I_{N}}\exp(P^{\prime}(Qn+r))\bigotimes_{i=k_{0}}^{k}T_{p_{i}(Qn+r)v_{i}}f_{i}\Biggr{\|}_{L^{2}(\mu^{\otimes
t})}=0,$

where $t=k-k_{0}+1$. Fix $r\in\\{0,\dots,Q-1\\}^{L}$ and set
$R(n)=P^{\prime}(Qn+r)$. Let $p\colon{\mathbb{Z}}^{L}\to{\mathbb{Z}}^{dt}$ be
the polynomial given by

$p(n)=(p_{i}(Qn+r)v_{i})_{k_{0}\leq i\leq k}.$

Let $(X^{t},\mathcal{B}^{t},\mu^{t},(S_{g})_{g\in{\mathbb{Z}}^{dt}})$ be the
${\mathbb{Z}}^{dt}$-system such that

$S_{(u_{i})_{k_{0}\leq i\leq k}}\coloneqq\prod_{i=k_{0}}^{k}T_{u_{i}}$

for all $u_{i}\in{\mathbb{Z}}^{d},k_{0}\leq i\leq k$, and denote
$f\coloneqq\bigotimes_{i=k_{0}}^{k}f_{i}$. We may then rewrite the left hand
side of (35) as

(36) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\|\mathbb{E}_{n\in
I_{N}}\exp(R(n))S_{p(n)}f\|_{L^{2}(\mu^{\otimes t})}.\end{split}$

For $K=\deg(p_{k_{0}})-1,$ using Lemma 4.2, to show that (36) is zero, it
suffices to show

(37)
$\begin{split}\overline{\mathbb{E}}^{\square}_{{\bf{h}}=(h_{1},\dots,h_{K})\in({\mathbb{Z}}^{L})^{K}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\|\mathbb{E}_{n\in
I_{N}}\exp(\Delta^{K}R(n,{\bf{h}}))S_{\Delta^{K}p(n,{\bf{h}})}f\|_{L^{2}(\mu^{\otimes
t})}=0.\end{split}$

By assumption, $\Delta^{K}R(n,h_{1},\dots,h_{K})$ is of degree 1 in the
variable $n$. Since $\deg(p)=\deg(p_{k_{0}})\geq\deg(p_{i})$ for all $i\geq
k_{0}$, $\Delta^{K}p(n,h_{1},\dots,h_{K})$ is also of degree 1 in the variable
$n$. We may thus assume that

$\Delta^{K}p(n,h_{1},\dots,h_{K})=((c_{i}(h_{1},\dots,h_{K})\cdot
n+c^{\prime}_{i}(h_{1},\dots,h_{K}))v_{i})_{k_{0}\leq i\leq k},$

for some polynomials
$c_{k_{0}},\dots,c_{k}\colon{\mathbb{Z}}^{LK}\to{\mathbb{Z}}^{L}$ and
$c^{\prime}_{k_{0}},\dots,c^{\prime}_{k}\colon{\mathbb{Z}}^{LK}\to{\mathbb{Z}}$.
Write
$\mathbb{c}(h_{1},\dots,h_{K})\coloneqq(c_{i}(h_{1},\dots,h_{K})v_{i})_{k_{0}\leq
i\leq k}$ (which is viewed as a $t$-tuple of $L$-tuple of vectors in
${\mathbb{Z}}^{d}$). If we write

$c_{i}(h_{1},\dots,h_{K})=(c_{i,1}(h_{1},\dots,h_{K}),\dots,c_{i,L}(h_{1},\dots,h_{K}))$

for some $c_{i,j}(h_{1},\dots,h_{K})\in{\mathbb{Z}}$, then, by definition,
$G(\mathbb{c}(h_{1},\dots,c_{K}))$ is the subgroup of ${\mathbb{Z}}^{dt}$
generated by the elements

$(c_{k_{0},j}(h_{1},\dots,h_{K})v_{k_{0}},\dots,c_{k,j}(h_{1},\dots,h_{K})v_{k}),1\leq
j\leq L.$

By Lemma 4.3, the left hand side of (37) is bounded by a constant multiple of

(38)
$\begin{split}(\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{K}\in{\mathbb{Z}}^{L}}\lvert\\!|\\!|f_{k_{0}}|\\!|\\!\rvert^{4}_{G(c_{k_{0}}(h_{1},\dots,c_{K})v_{k_{0}})^{\times
2}})^{1/4},\end{split}$

where $G(c_{k_{0}}(h_{1},\dots,c_{K})v_{k_{0}})$ is the subgroup of
${\mathbb{Z}}^{d}$ generated by the elements

$c_{k_{0},1}(h_{1},\dots,h_{K})v_{k_{0}},\dots,c_{k_{0},L}(h_{1},\dots,h_{K})v_{k_{0}},$

i.e., the entries of $c_{k_{0}}(h_{1},\dots,h_{K})v_{k_{0}}$. For any
$u_{k_{0}}\in G(c_{k_{0}}(h_{1},\dots,h_{K})v_{k_{0}})$, note that $u_{k_{0}}$
is a rational multiple of $v_{k_{0}}$. So, if
$c_{k_{0}}(h_{1},\dots,h_{K})\neq\mathbb{0}$, then
$G(c_{k_{0}}(h_{1},\dots,h_{K})v_{k_{0}})=G(v_{k_{0}})$.

Since $(T_{p_{i}(n)v_{i}})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu$, we
have that $T_{v_{i}}$ is ergodic for $\mu$. As
$\mathbb{E}(f_{k_{0}}|Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}(\mathbb{X}))=0$,
by [6, Lemma 2.4], we have that

$\lvert\\!|\\!|f_{k_{0}}|\\!|\\!\rvert_{G(c_{k_{0}}(h_{1},\dots,c_{K})v_{k_{0}})^{\times
2}}=\lvert\\!|\\!|f_{k_{0}}|\\!|\\!\rvert_{v_{k_{0}}^{\times
2}}=\lvert\\!|\\!|f_{k_{0}}|\\!|\\!\rvert_{({\mathbb{Z}}^{d})^{\times 2}}=0$

whenever $c_{k_{0}}(h_{1},\dots,h_{K})\neq\mathbb{0}$. Since
$K=\deg(p_{k_{0}})-1=\deg(p_{k_{0}}(Q\cdot+r))-1$, it is easy to see that
$c_{k_{0}}\not\equiv\mathbb{0}$. By [6, Lemma 2.12], the set of such
$(h_{1},\dots,h_{K})$ is of density 1. So, averaging over all
$h_{1},\dots,h_{K}\in{\mathbb{Z}}^{L}$, we have that (38) is 0. This finishes
the proof of the claim.

Using the claim, it suffices to prove (31) under the assumption that all
$f_{i}$’s are measurable with respect to $Z_{({\mathbb{Z}}^{d})^{\times
2}}(\mathbb{X})$. By Lemma 2.7 of [6], we can approximate each $f_{i}$ in
$L^{2}(\mu)$ by an eigenfunction of $\mathbb{X}$. By multi-linearity, we may
assume without loss of generality that each $f_{i}$ is a non-constant
eigenfunction of $\mathbb{X}$ given by $T_{n}f_{i}=\exp(\lambda_{i}(n))f_{i}$
for all $n\in\mathbb{Z}^{d}$ for some group homomorphism
$\lambda_{i}\colon\mathbb{Z}^{d}\to{\mathbb{R}},$ and that $f_{i}(x)\neq 0$
$\mu$-a.e $x\in X$. Then, since $(T_{p_{i}(n)v_{i}}\colon 1\leq i\leq
k)_{n\in{\mathbb{Z}}^{L}}$ is jointly ergodic for $\mu$, for any Følner
sequence $(I_{N})_{N\in{\mathbb{N}}}$ of ${\mathbb{Z}}^{d}$,

$0=\prod_{i=1}^{k}\int_{X}f_{i}d\mu=\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}T_{p_{i}(n)v_{i}}f_{i}=\left(\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}\exp(\lambda_{i}(p_{i}(n)v_{i}))\right)\prod_{i=1}^{k}f_{i}.$

This implies that $\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}\exp(\lambda_{i}(p_{i}(n)v_{i}))=0$. So,

$\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}\bigotimes_{i=1}^{k}T_{p_{i}(n)v_{i}}f_{i}=\left(\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}\exp(\lambda_{i}(p_{i}(n)v_{i}))\right)\bigotimes_{i=1}^{k}f_{i}.$

This finishes the proof. ∎

We are now ready to complete the proof of Theorem 2.3.

###### Proof of Theorem 2.3.

Using Proposition 6.2, we have that (C1) implies (C2). It is obvious that (C2)
implies (C2’). So, it suffices to show that (C2’) implies (C1).

It is not hard to see that we may assume without loss of generality that
$p_{i}(0)=0$ for all $1\leq i\leq k$. By Theorem 2.4, to show that
$(T_{p_{i}(n)v_{i}}\colon 1\leq i\leq k)_{n\in{\mathbb{Z}}^{L}}$ is jointly
ergodic for $\mu$, it suffices to show that $G_{i,j}(\mathbb{p})$ is ergodic
for $\mu$ for all $0\leq i,j\leq k,i\neq j$. Fix any such pair $(i,j)$. We may
assume without loss of generality that $i\neq 0$. If $j=0$, then by (ii),
$(T_{p_{i}(n)v_{i}})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu$. So
$T_{v_{i}}$ is ergodic for $\mu$, and thus $G_{i,0}(\mathbb{p})=G(v_{i})$ is
ergodic for $\mu$. So, we may now assume that $j\neq 0$.

Assume first that $\deg(p_{i})=\deg(p_{j})$. By assumption, either $v_{i}$ and
$v_{j}$ are linearly dependent, or $p_{i}(n)$ and $p_{j}(n)$ are linearly
dependent.

If $v_{i}$ and $v_{j}$ are linearly dependent over ${\mathbb{Z}}$, then we may
assume without loss of generality that $v_{i}=av$ and $v_{j}=bv$ for some
$a,b\in\mathbb{Q}$ and $v\in{\mathbb{Z}}^{d}$. By (i),
$(T_{(ap_{i}(n)-bp_{j}(n))v})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu$,
which implies that $G(v)$ is ergodic for $\mu$. On the other hand,
$G_{i,j}(\mathbb{p})$ is a group generated by some elements which are linear
combinations of $v_{i}$ and $v_{j}$, which are thus multiples of $v$. Since
$\mathbb{p}$ is non-degenerate, $G_{i,j}(\mathbb{p})$ is not the trivial
group. It follows that $G_{i,j}(\mathbb{p})=G(v),$ so the group
$G_{i,j}(\mathbb{p})$ is ergodic for $\mu$.

If $p_{i}(n)$ and $p_{j}(n)$ are linearly dependent over ${\mathbb{Z}}$, then
we may assume without loss of generality that $p_{i}(n)=ap(n)$ and
$p_{j}(n)=bp(n)$ for some $a,b\in\mathbb{Q}$ and polynomial $p$. By (i),
$(T_{(p(n)(av_{i}-bv_{j})})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for $\mu$,
which implies that $G(av_{i}-bv_{j})$ is ergodic for $\mu$. On the other hand,
$G_{i,j}(\mathbb{p})$ is a group generated by some elements which are
multiples of $av_{i}-bv_{j}$. Since $\mathbb{p}$ is non-degenerate,
$G_{i,j}(\mathbb{p})$ is not the trivial group. It follows that
$G_{i,j}(\mathbb{p})=G(av_{i}-bv_{j}),$ so the group $G_{i,j}(\mathbb{p})$ is
ergodic for $\mu$.

Finally, we consider the case when $\deg(p_{i})\neq\deg(p_{j})$. We may assume
without loss of generality that $\deg(p_{i})>\deg(p_{j})$. In this case,
$G_{i,j}(\mathbb{p})=G_{i,0}(\mathbb{p})$, which we have shown is ergodic for
$\mu$. ∎

##  7\. Potential future directions

We close this article with two potential future directions regarding the
splitting of multicorrelation sequences. The first one is for integer
polynomial iterates under no assumptions on the transformations other than
commutativity (see Theorem 7.1 for a special case of two terms).

The second one pertains to potential results analogous to Theorem 2.2 for
iterates of the form $[p_{i}(n)],$ $1\leq i\leq k,$ where
$p_{i}=(p_{i,1},\ldots,p_{i,d}):\mathbb{Z}^{L}\to\mathbb{R}^{d}$ are vectors
of real valued polynomials.353535 Here, for
$x=(x_{1},\ldots,x_{L})\in\mathbb{R}^{L},$ we write
$[x]\coloneqq([x_{1}],\ldots,[x_{L}]),$ where $[\cdot]$ is the floor function.
In fact, one can consider any combination of rounding functions, i.e., floor,
ceiling, or closest integer.

###  7.1. The two-term case with no ergodicity assumptions

Given the results in the appendix of [5], we are able to obtain the following
splitting result for two commuting transformations without any ergodicity
assumptions:

######  Theorem 7.1.

Let $(X,\mathcal{B},\mu,T,S)$ be a measure preserving system with $TS=ST$. Let
$f_{0},f_{1},f_{2}\in L^{\infty}(\mu)$ and $p\in{\mathbb{Z}}[n]$ with degree
$K\geq 2$. Then, the multicorrelation sequence

$a(n)\coloneqq\int_{X}f_{0}\cdot T^{n}f_{1}\cdot S^{p(n)}f_{2}\ d\mu$

can be decomposed as a sum of a uniform limit of $K$-step nilsequences plus a
nullsequence.

###### Proof.

Setting $F_{i}=f_{i}\otimes\bar{f}_{i},$ $i=0,1,2,$ and
$\tilde{\mu}=\mu\times\mu,$ we have that

(39)
$\frac{1}{N}\sum_{n=1}^{N}|a(n)|^{2}=\frac{1}{N}\sum_{n=1}^{N}\int_{X^{2}}F_{0}\cdot(T\times
T)^{n}F_{1}\cdot(S\times S)^{p(n)}F_{2}\ d\tilde{\mu}.$

Using [5, Theorem A.3], we get that the rational Kronecker factor363636The
_rational Kronecker factor_ is the smallest sub-$\sigma$-algebra of
$\mathcal{B}$ that makes all functions with finite orbit in $L^{2}(\mu)$ under
the transformation $T$ measurable. is characteristic for the averages
appearing in (39). Consequently, we may replace $f_{1}$ by $P_{c}f_{1}$ and
$f_{2}$ by $Q_{c}f_{2}$ in $a(n)$ up to a nullsequence, where $P_{c}$ denotes
the orthogonal projection onto the compact component of the splitting
associated to $T$, and $Q_{c}$ the one associated to $S$. (Here we make use of
the Hilbert space splitting of $L^{2}(\mu)$ into its compact and weakly mixing
components for a given unitary operator. The seeds for these results are
already present in the work of Koopman and von Neumann [21]. They were later
generalized by Jacobs, Glicksberg and de Leeuw. See [7, Section 16.3] for a
more modern treatment). Thus, the sequence

$a(n)-\int_{X}f_{0}\cdot T^{n}P_{c}f_{1}\cdot S^{p(n)}Q_{c}f_{2}\ d\mu$

is a nullsequence. Let $\varepsilon>0$ and choose
$h_{1},\dots,h_{k},g_{1},\dots,g_{k}\in L^{2}(\mu)$ such that
$Th_{i}=\lambda_{i}h_{i}$ and $Sg_{i}=\rho_{i}g_{i}$ (for some
$\lambda_{1},\dots,\lambda_{k},\rho_{1},\dots,\rho_{k}\in{\mathbb{C}}$ of
absolute value one) as well as
$a_{1},\dots,a_{k},b_{1},\dots,b_{k}\in{\mathbb{C}}$ such that

$\left|\int_{X}f_{0}\cdot T^{n}P_{c}f_{1}\cdot S^{p(n)}Q_{c}f_{2}\
d\mu-\int_{X}f_{0}\cdot T^{n}\sum_{i=1}^{k}a_{i}h_{i}\cdot
S^{p(n)}\sum_{j=1}^{k}b_{j}g_{j}\ d\mu\right|<\varepsilon.$

Observe that

$\displaystyle\int_{X}f_{0}\cdot T^{n}\sum_{i=1}^{k}a_{i}h_{i}\cdot
S^{p(n)}\sum_{j=1}^{k}b_{i}g_{i}\ d\mu$ $\displaystyle=$
$\displaystyle\int_{X}f_{0}\cdot\sum_{i=1}^{k}a_{i}\lambda_{i}^{n}h_{i}\cdot\sum_{j=1}^{k}b_{j}\rho_{j}^{p(n)}g_{j}\
d\mu$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{k}\left(a_{i}b_{j}\int_{X}f_{0}\cdot h_{i}\cdot
g_{j}\ d\mu\right)\lambda_{i}^{n}\rho_{j}^{p(n)},$

which is a $K$-step nilsequence. Applying the same argument as in the proof
Theorem 2.2 we deduce the decomposition result. The rest of the details are
omitted for the sake of brevity. ∎

It is natural to ask whether a result analogous to Theorem 7.1 holds for
longer expressions (potentially via a generalization of the results in the
appendix of [5]), and with more general polynomial iterates, even without
necessarily assuming they have distinct degrees. Thus, we state the following
problem:

######  Problem 1.

Obtain decomposition results of the form “uniform limit of nilsequences plus a
nullsequence” for multicorrelation sequences with (integer) polynomial
iterates for general systems under no ergodicity assumptions on the
transformations.

###  7.2. Integer part polynomial iterates

With a, by now, standard argument (introduced in [4] and [29] for a single
term, extended for two terms in [34], and further developed in [22], [23] and
[25]) one has, for the vectors of real polynomials
$p_{i}=(p_{i,1},\ldots,p_{i,d}),$ that the expression

(40) $\frac{1}{|I_{N}|}\sum_{n\in
I_{N}}\prod_{i=1}^{k}T_{[p_{i}(n)]}f_{i}=\frac{1}{|I_{N}|}\sum_{n\in
I_{N}}\prod_{i=1}^{k}\prod_{j=1}^{d}T_{j}^{[p_{i,j}(n)]}f_{i}$

is “close” to

(41) $\frac{1}{|I_{N}|}\sum_{n\in
I_{N}}\prod_{i=1}^{k}\prod_{j=1}^{d}S_{j}^{p_{i,j}(n)}g_{i},$

where $S_{j}$’s are $\mathbb{R}$-flows on an “extension system” $Y$ of $X$,
and the functions $g_{i}$’s are extensions of the $f_{i}$’s (see [23] for
details).373737 We say that a jointly measurable family
$(S_{t})_{t\in{\mathbb{R}}^{d}}$ of measure preserving transformations on a
probability space, is an _${\mathbb{R}}^{d}$ -action_ (_flow_), if it
satisfies $S_{t+r}=S_{t}\circ S_{r}$ for all $t,r\in{\mathbb{R}}^{d}.$ As an
application of Theorem 2.2, one can prove splitting theorems for
${\mathbb{R}}^{d}$-actions on the extension system.

Indeed,383838 We address the $L=1$ case for simplicity; following the same
argument one can similarly get the corresponding result for the general case
of $L$-variable polynomials by using an ordering on the parameters, e.g.,
$n_{1}>\ldots>n_{L}.$ consider the multicorrelation sequence

(42) $\int_{X}f_{0}\cdot S_{p_{1}(n)}f_{1}\cdot\dotso\cdot S_{p_{k}(n)}f_{k}\
d\mu,$

where $S$ is a measure preserving ${\mathbb{R}}^{d}$-action on the probability
space $(X,\mathcal{B},\mu),$ $f_{0},f_{1},\ldots,f_{k}\in L^{\infty}(X),$ and
$p_{1},\dots,p_{k}\colon\mathbb{Z}\to\mathbb{R}^{d}$ a non-degenerate family
of polynomials of degree at most $K$ with
$p_{i}(n)=\sum_{h=0}^{K}a_{i,h}n^{h}$, $a_{i,h}\in\mathbb{R}^{d}$. Then

$S_{p_{i}(n)}=S_{\sum_{h=0}^{K}a_{i,h}n^{h}}=\prod_{h=0}^{K}(S_{a_{i,h}})^{n^{h}}.$

Note that $S_{a_{i,h}},1\leq i\leq k,1\leq h\leq K$ generate a
${\mathbb{Z}}^{kK}$-action on $(X,\mathcal{B},\mu)$. For convenience, set
$p_{0}$ to be the constant zero polynomial. For $0\leq i,j\leq k,i\neq j$, let
$D_{i,j}$ be the largest integer $h$ such that
$S_{a_{i,D_{i,j}}-a_{j,D_{i,j}}}\neq\textrm{id}$. This transformation will be
denoted by $R_{i,j}$. By Theorem 2.2, one can show that the desired splitting
result for the sequence (42), if all the transformations $R_{i,j},0\leq
i,j\leq k,i\neq j$ are all ergodic (as ${\mathbb{Z}}$-actions on the extension
system $Y$).

Unfortunately, even though we have the previous result for flows, the error
term that arises from the approximation of (40) by (41) prevents us from
getting the conclusion of Theorem 2.2 for multicorrelation sequences of the
form

$\int_{X}f_{0}\cdot T_{[p_{1}(n)]}f_{1}\cdot\dotso\cdot T_{[p_{k}(n)]}f_{k}\
d\mu.$

######  Remark 7.2.

It is important to stress that, for integer part real polynomial iterates, one
does not expect to have the desired multicorrelation splitting in general. The
next example shows that, even for $k=1,$ an ergodic system, and linear
iterates, it can be too much to hope for:

_Indeed, following[25, Example 7], let
$X=\mathbb{T}\coloneqq\mathbb{R}/\mathbb{Z}$, $T(x)=x+1/\sqrt{2}$,
$p(n)=\sqrt{2}n$, $f_{0}(x)=e(x)$ and $f_{1}(x)=e(-x)$, where $e(x)\coloneqq
e^{2\pi ix}$. Then, we have that_

$\displaystyle\int f_{0}\cdot T^{[p(n)]}f_{1}\,d\mu$ $\displaystyle=$
$\displaystyle\int
e(x)e\left(-x-\frac{1}{\sqrt{2}}[\sqrt{2}n]\right)\,dx=e\left(-\frac{1}{\sqrt{2}}[\sqrt{2}n]\right)=e\left(\frac{1}{\sqrt{2}}\\{\sqrt{2}n\\}\right),$

_which cannot be written as a uniform limit of nilsequences and a
nullsequence._

######  Remark 7.3.

One may think that the fact that $\sqrt{2}$ and $\frac{1}{\sqrt{2}}$ are not
linearly independent over ${\mathbb{Q}}$ is behind the impossibility of the
splitting in the example above. However, a closer examination of the proof
given in [25] shows that this is not the case, and that the failure extends
quite generally.

_Indeed, we can imitate the example quoted above as follows. Let
$(X,\mathcal{B},\mu,T)$ be an ergodic measure preserving system with non-
trivial irrational spectrum. Let $f_{1}:X\to\mathbb{S}^{1}$ be an
eigenfunction of $T$ with eigenvalue $e^{2\pi i\beta}$, with
$\beta\in{\mathbb{R}}\setminus{\mathbb{Q}}$. Put $f_{0}=\bar{f_{1}}$. Then, if
we consider the multicorrelation sequence_

$a(n)\coloneqq\int_{X}f_{0}\cdot T^{[\alpha n]}f_{1}\ d\mu,$

_with the choices made above, we observe that, in fact, $a(n)=e^{2\pi i[\alpha
n]\beta}.$ The same argument as in [25, Example 7] shows that $a(n)$ cannot be
written as a uniform limit of nilsequences plus a nullsequence._

On the other hand, if we postulate very strong assumptions on our
transformations, we do have the desired decomposition results. For example
(see [23]), if $T_{1},\ldots,T_{k}$ are commuting weakly mixing
transformations on $(X,\mathcal{B},\mu)$, $q_{i}(n)=p_{i}(n)e_{i},$ $1\leq
i\leq k,$ where $p_{i}\colon\mathbb{Z}\to\mathbb{R}$ are real polynomials of
distinct, positive degrees, and $f_{1},\dots,f_{k}\in L^{\infty}(\mu),$ then
we have that

$\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}T_{[q_{1}(n)]}f_{1}\cdot\dotso\cdot
T_{[q_{k}(n)]}f_{k}=\prod_{i=1}^{k}\int_{X}f_{i}\ d\mu.$

Hence, for any $f_{0}\in L^{\infty}(\mu),$ the multicorrelation sequence

$\int_{X}f_{0}\cdot T_{[q_{1}(n)]}f_{1}\cdot\dotso\cdot T_{[q_{k}(n)]}f_{k}\
d\mu$

can be written as a sum of a constant (i.e., a $0$-step nilsequence) and a
nullsequence.

We conclude this article with the following problem that arises naturally:

######  Problem 2.

Let $d,k,K,L\in\mathbb{N},$
$p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{R}^{d}$ be a non-degenerate
family of polynomials of degree at most $K$,
$(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ a measure preserving system and
$f_{0},\dots,f_{k}\in L^{\infty}(\mu).$ Find conditions, on the $p_{i}$’s
and/or the $\mathbb{Z}^{d}$-action $T$ that is defined by the $T_{i}$’s, so
that the multicorrelation sequence

$\int_{X}f_{0}\cdot T_{[p_{1}(n)]}f_{1}\cdot\ldots\cdot T_{[p_{k}(n)]}f_{k}\
d\mu$

can be decomposed as a sum of a uniform limit of $D$-step nilsequences and a
nullsequence.

## References

  * [1] D. Berend and V. Bergelson. Jointly ergodic measure-preserving transformations. _Israel J. Math._ 49 (1984), no. 4, 307–314.
  * [2] V. Bergelson. Weakly mixing PET. _Ergodic Theory Dynam. Systems_ 7 (1987), no. 3, 337–349.
  * [3] V. Bergelson, B. Host and B. Kra, with an appendix by I. Ruzsa. Multiple recurrence and nilsequences, _Invent. Math._ 160 (2005), no. 2, 261–303.
  * [4] M. Boshernitzan, R. L. Jones and M. Wierdl. Integer and fractional parts of good averaging sequences in ergodic theory. Convergence in Ergodic Theory and Probability, Eds: V. Bergelson, P. March, J. Rosenblatt, by Walter de Gruyter & Co., Berlin, New York 1996, 117–132.
  * [5] Q. Chu, N. Frantzikinakis and B. Host. Ergodic averages of commuting transformations with distinct degree polynomial iterates. _Proc. Lond. Math. Soc._ (3) 102 (2011), no. 5, 801–842.
  * [6] S. Donoso, A. Koutsogiannis and W. Sun. Seminorms for multiple averages along polynomials and applications to joint ergodicity. To appear in _J. d’Analyse Mathématique_.
  * [7] T. Eisner, B. Farkas, M. Haase and R. Nagel. Operator theoretic aspects of ergodic theory, Graduate Texts in Mathematics, 272, Springer, Cham, 2015.
  * [8] A. F. Moragues, Properties of multicorrelation sequences and large returns under some ergodicity assumptions, Discrete Contin. Dyn. Syst. 41 (2021), no. 6, 2809–2828.
  * [9] N. Frantzikinakis. Joint ergodicity of sequences. Preprint 2021, arXiv:2102.09967.
  * [10] N. Frantzikinakis, B. Host and B. Kra. The polynomial multidimensional Szemerédi theorem along shifted primes. Israel J. Math. 194 (2013), no. 1, 331–348.
  * [11] N. Frantzikinakis. Multiple correlation sequences and nilsequences. _Invent. Math._ 202 (2015), no. 2, 875–892.
  * [12] N. Frantzikinakis and B. Host. Weighted multiple ergodic averages and correlation sequences, _Ergodic Theory Dynam. Systems_ 38 (2018), no. 1, 81–142.
  * [13] N. Frantzikinakis and B. Kra. Polynomial averages converge to the product of integrals. _Israel J. Math._ 148 (2005), 267–276.
  * [14] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. _J. d’Analyse Mathématique_ 31 (1977), 204–256.
  * [15] B. Host. Ergodic seminorms for commuting transformations and applications. _Studia Math._ 195 (2009), no. 1, 31–49.
  * [16] B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. _Ann. of Math._ (2) 161 (2005), no. 1, 397–488.
  * [17] B. Host and B. Kra. Uniformity seminorms on $\ell^{\infty}$ and applications, _J. Anal. Math._ 108 (2009), 219–276.
  * [18] M. C. R. Johnson. Convergence of polynomial ergodic averages of several variables for some commuting transformations. _Illinois J. Math._ 53 (2009), no. 3, 865–882 (2010).
  * [19] D. Karageorgos and A. Koutsogiannis. Integer part independent polynomial averages and applications along primes.Studia Mathematica 249 (2019), no. 3, 233–257.
  * [20] A. Khintchine. The Method of Spectral Reduction in Classical Dynamics. _Proceedings of the National Academy of Sciences_ May 1933, 19 (5) 567-573.
  * [21] B. O. Koopman and J. von Neumann. Dynamical Systems of Continuous Spectra. Proceedings of the National Academy of Sciences Mar 1932, 18 (3) 255-263.
  * [22] A. Koutsogiannis. Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems 38 (2018), no. 2, 666–685.
  * [23] A. Koutsogiannis. Integer part polynomial correlation sequences. Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1525–1542.
  * [24] A. Koutsogiannis. Multiple ergodic averages for variable polynomials. Preprint 2021, arXiv:2101.00534.
  * [25] A. Koutsogiannis, A. Le, J. Moreira and F. K. Richter. Structure of multicorrelation sequences with integer part polynomial iterates along the primes. Proceedings of the A. M. S. 149 (2021), No 1, 209–216.
  * [26] A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. _Israel J. Math._ 146 (2005), 303–315.
  * [27] A. Leibman. Multiple polynomial correlation sequences and nilsequences. Ergodic Theory Dynam. Systems 30 (2010), no. 3, 841–854.
  * [28] A. Leibman. Nilsequences, null-sequences, and multiple correlation sequences. Ergodic Theory Dynam. Systems 35 (2015), no. 1, 176–191.
  * [29] E. Lesigne. On the sequence of integer parts of a good sequence for the ergodic theorem. Comment. Math. Univ. Carolin. 36 (1995), no. 4, 737–743.
  * [30] W. Sun. Weak ergodic averages over dilated curves. _Ergodic Theory Dynam. Systems_ 41 (2021), No 2, 606–621.
  * [31] T. Tao and J. Teräväinen. The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures. _Duke Math. J._ , 168 (2019), No 11, 1977–2027.
  * [32] T. Tao and T. Ziegler. Concatenation Theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors. _Discrete Anal._ 2016, Paper No. 13.
  * [33] M. Walsh. Norm convergence of nilpotent ergodic averages. _Ann. of Math._ (2) 175 (2012), no. 3, 1667–1688.
  * [34] M. Wierdl. Personal communication with the third author (2015).
  * [35] T. Ziegler. Nilfactors of $\mathbb{R}^{m}$-actions and configurations in sets of positive upper density in $\mathbb{R}^{m}$. _J. d’Analyse Mathématique_ 99 (2006), 249–266.