# Surface Energy Driven Grain Growth Model: FePt L10 Nanoparticles

Connor Skelland  College of Engineering, Mathematics, and Physical Sciences,
University of Exeter, Exeter, Devon, United Kingdom
<EMAIL_ADDRESS>Gino Hrkac  College of Engineering,
Mathematics, and Physical Sciences, University of Exeter, Exeter, Devon,
United Kingdom

###### Abstract

We developed a grain growth model that is based on the energy minimisation of
surfaces with respect to the volume energy and the grain’s environment. We
used the well known FePt L1${}_{\text{0}}$ system to discover the physical
factors that drive the shape and size of FePt grains. It was found that the
preferred growth directions are along symmetry planes that are determined by
the basic crystal and driven by surface energy minimisation. The model
developed here can be used to predict a grain’s growth and shape as a function
of atomic number and composition. This means that by tailoring a grain’s
surface and grain boundaries the shape of the magnetic grains can be
manipulated.

## Introduction

Magnetic materials have a long history of driving technological growth and
societal progress[1]. In the modern world, three of the most important
industries, Green-Energy, the Automotive-Sector, and Magnetic Recording are
reliant on the improvement of at least one of a magnetic materials fundamental
properties - coercivity, anisotropy, stability, etc. Throughout history, the
greatest leaps in improvement have come from step changes in performance
caused by the discovery of impressive new crystal structures. For example, the
transition from Steel to Alnico, and Alnico to Rare-Earth based magnetic
materials[2].

However, since the discovery of Nd2Fe14B in 1984[3], the hard permanent magnet
research community has struggled to find new crystal structures with improved
magnetic properties and a compelling price point. In the intervening time, the
focus has shifted to optimising material properties via microstructure
engineering, which is achieved through alterations to fabrication process and
minor alterations to chemical composition[4, 5, 6, 7, 8, 9]. The success of
this method is exemplified by the anistropic sintered Nd2Fe14B magnets,
detailed in a patent by Sakuma et al.[10].

The success of microstructure engineering demonstrates the importance of
controlling magnetism through structural control. The implication of this is
that physical models must capture defects, phase inhomogeneities, grain shape,
grain size, and grain orientation if they are to reproduce realistic magnetic
effects.

Currently microstructure informed modelling is limited - a combination of
Voronoi tesselation and integranular phase definition is the most accurate
template used for both micromagnetic and spin dynamic modelling [11, 12, 13,
14, 15]. Whilst capable of capturing the general size and rough shape of
grains, these models are limited by common simplifications such as:
intergranular phases are indistinguishable in their magnetic properties, all
intergranular phases are the same width, and multi-phase grains share average
values in anisotropy and exchange. Although reasonable, these simplifications
prevent the models from totally capturing grain boundary effects that cause
domain wall pinning, reverse grain nucleation, and intergranular exchange.
These effects have a prominent role in two of the most important magnetic
crystal structures: Nd2Fe14B[16, 7] and FePt L10[17] and therefore must be
taken into consideration. To do this, models must accurately represent
microstructure.

To address this problem, we took the view that accurate representation of
microstructure starts with an understanding of what drives its development.
This led us to produce a grain growth model that let us investigate how
surface energy defines preferential grain shapes, and how preferential shapes
change with grain size. We applied this model to FePt L10 for two reasons,
firstly, it is a well studied structure, and secondly, grain size and shape
are crucial to its application in the magnetic hard drive industry[18, 19].
Particularly, grains homogeneous in size and shape improve the signal to noise
ratio of hard drive recording tracks[20, 21], decreasing the number of grains
required per bit and increasing information density as a result.

Our model uses Molecular-Dynamics-based energy minimisation to calculate the
preferability of grains composed of identical elements; their identical
composition makes the grains different states of the same thermodynamic
system. Using the Boltzmann distribution of the underlying system, we assigned
an energy based probability to each of the grains (which are considered states
within the same Boltzmann distribution), which made it possible to establish
the relative probability of each type of grain - see the methodology section.
This is a first step towards a general methodology for producing realistic
atomistic models, that can be used as input to Heisenberg spin models and
finite element based micromagnetic modelling. Future applications of this
model will include the investigation of granular rare-earth intermetallics for
the automotive industry.

## Results

(a)

(b)

(c)

Figure 1: Compositionally matched grain morhpologies at the size of 10,000
atoms. Showing a) Octahedron, b) Truncated Octahedron Major, and c) Truncated
Octahedron Minor. The arrows represent the grain’s surface normals.

To produce realistic grain shapes, we calculated the surface energies of the
five lowest index symmetry planes of FePt L10 in LAMMPS, the results of this
can be seen in Table 1 along with a comparison to calculations by Kim et
al.[22]. We used these surface energies to generate five types of grain shape
across a range of sizes.

Plane | Our Surface Energy (erg/cm2) | Kim’s Surface Energy (erg/cm2)
---|---|---
111 | 2200 | 2198
101 | 2685 | 2714
110 | 2756 | 2650
100 | 2449 | 2719
001 | 2919 | 2740
Table 1: Surface Energies of low index FePt L1${}_{\text{0}}$ planes, taken
from Kim et al.[22], compared to our own calculated values.

As the table shows, the {111} surfaces have the lowest energy, informing our
choice for three of the grain shapes, which can be seen in Figure 1. As the
Figure shows, these shapes are defined so that they feature predominantly
{111} surfaces, which our surface calculations predict would make them the
lowest energy structures. As a comparison, we included two extra shapes, which
can be seen in Figure 2. We chose a spherical and cuboid grain for their
surface area to volume ratios. Selecting shapes at the extreme ends of the
surface area to volume scale, allowed us to investigate what role surface area
to volume plays in energetic preferability.

(a)

(b)

Figure 2: Compositionally matched grain morhpologies at the size of 10,000
atoms. Showing a) Cuboid, and b) Sphere. Arrows on the cuboid show surface
normal directions, arrows are not included on the sphere to avoid clutter.

From Figure 1, the grain shapes are named a) Octahedron, b) Truncated-
Octahedron-Major, and c) Truncated-Octahedron-Minor; from Figure 2, the grains
are named a) Cuboid, b) Sphere.

To compare the preferability of these shapes as a function of size, we created
grains of each type ranging from 1,000 to 15,000 atoms, in steps of 1,000
atoms - equating to diameters of 3-9nm. To ensure our grains had identical
compositions, we scaled the grain structures above their intended atom target,
then deleted atoms from their surface to match an average composition with the
desired atom number - for full details see the methodology section. An example
of how this affects the structure can be seen in Figure 3, which shows how the
surface of the grain structure changes to reach a matching composition. We
refer to this process as composition matching, and the grains in this state as
"as-matched".

(a)

(b)

Figure 3: Comparison of a 20,000 atom Truncated Octahedron Minor in its a) As
built, and b) Compositionally matched, states. Figure 4: Boltzmann Factors for
each structure over a range of 1,000 - 15,000 atoms.

We calculated the energy of our as-matched grains using conjugate gradient
energy minimisation at 0K[23]. As mentioned, the grains have identical
compositions, making them re-arrangements of the same underlying system. This
allowed us to calculate their relative probabilities in the Boltzmann
distribution by calculating their Boltzmann factors from their minimised
energy. The ratio of their Boltzmann factors gave us their comparative
probability with temperature - see methodology section for full details.

In Figure 4, we show that, across the investigated size range, the most
probable shapes are alternately the Octahedron, and the Truncated-Octahedron-
Minor. This demonstrates the competition between beneficial surface selection
and surface area to volume ratio, as their similar energies are due to the
interplay of these two factors.

In the case of the Octahedron, on average, the surface is 87% {111} planes,
giving it the lowest surface energy per unit area of the all grain shapes.
However, as a result of maximising the {111} surface, this grain shape has the
worst surface area to volume ratio, giving it substantially more surface for
the same volume. In contrast, the Truncated-Octahedron-Minor has an average of
71% {111} surface planes, with the best surface area to volume ratio out of
all the shapes - owing to the truncation of the {111} plane corners. The fact
these two structures have similar energy, demonstrates two things, firstly,
that, at these sizes, the dominant factor is surface type; secondly, that when
two grains have similar percentages of the same surface type, the deciding
factor is surface area to volume ratio.

For the remaining shapes, the order of preference is Sphere/Trunacted-
Octahedron-Major followed by Cuboid, where Sphere and Truncated-Octahedron-
Major can be considered interchangeable. In contrast to the previous pair, the
Sphere and Truncated-Octahedron-Major have comparable energies due to their
energetically similar surface types only. For Truncated-Octahedron-Major, this
means a surface composed of, on average, 46% {111} surface planes, with {100}
and {001} surface planes making up the remaining surface area; for the Sphere,
this means a surface composed of, on average, 53% {111} planes, with a wide
spread of surface planes making up the remaining surface area. The spread of
the Sphere’s remaining planes includes multiple high index, high energy planes
that give the Sphere an overall surface energy per unit area of similar value
to the Truncated-Octahedron-Major’s, despite it’s 7% advantage in {111}
surface plane coverage. The fact these two structures are alike, in both
surface area and volume, further supports our conclusion that, at these length
scales, surface type is the dominant factor controlling grain shape.

As expected, the Cuboid is significantly less likely than every other shape,
possessing no {111} surface planes, except those incidentally caused by
surface deletion, and a poor surface area to volume ratio.

To demonstrate the predominant role of surface type, we calculated the surface
energy for each of the grains across the size range. Figure 5 shows these
energies, normalised by the minimum surface energy at every size. As can be
seen, the preference order we outlined above is reflected in the surface
energies. For grains of identical composition, surface energy is the singular
defining factor and for grains of this size range, surface type is the
dominant factor affecting surface energy.

Figure 5: Normalised surface energies of the investigated grains across the
range 1,000-15,000 atoms. The normalisation is different at each atom size,
with the normalising value being the grain with the lowest surface energy at
each point.

## Discussion

We have developed a model which uses crystal symmetries to predict grain
growth and shape as a function of size - measured in total atoms. The
methodology we have developed is applicable to any crystal structure, and we
anticipate using it in the future to study the probable shapes of Rare-Earth
Intermetallics, particularly RT12 phases, whose stability, and therefore
manufacturability, is dependent on the microstructure surrounding its grains.

Magnetron-sputtering-based gas-phase condensation experiments, by Qui and
Wang[24], show FePt L10 has a strong preference for the Octahedron, and
Truncated-Octahedron-Minor shapes, verifying our model and methodology.

Further to this, the surface energy basis of our model allows us to make
predictions about the microstructure of as-annealed FePt L10 thin films, which
we predict will organise into FePt L10 configurations that maximise the {111}
plane. As the underlayer that supports the FePt L10 thin film, pre-disposes
the crystal structure to align its {001} direction perpendicular to the plane,
for isolated FePt the reordering will cause the as-deposited material to
conglomerate into hill like structures, whose slopes are composed of {111}
surface planes. A review of the literature shows that this does occur[25],
which reinforces the validity of our model and opens the possibility that
experimentalists and modellers will be able to produce quantitative data on
the physical reasoning behind the texturing process.

With development, the model will evolve from a single methodology into a suite
of tools, which we are building into a Python package: Grain Modeller. We
believe our package has two benefits, firstly, theoreticians and modellers can
use it to calculate realistic grain shapes and reduce assumptions in their
work; secondly, experimentalists can use it to create models that test grain
shape preferability under a range of conditions, allowing them to fine tune
their apparatus for a greater degree of structural control, before undertaking
an experiment.

We believe that this extension to the material science toolbox will increase
our communities collective ability to control microstructure and increase the
speed of magnetic material development.

## Methods

### Grain Creation

To create FePt L1${}_{\text{0}}$ grains, we produced a Python package, for
defining grain shapes, creating grain shapes, matching grain compositions, and
creating LAMMPS data and input files.

#### Grain Shape Definition

To make grain shapes reproducible over a wide range of sizes, we developed a
method of defining them abstractly, using a unitcell, a set of cuts, and a
repeat ratio. For clarity, these are defined below:

  * •

Unitcells, are the basis of the grain, defined by three lattice vectors
$\boldsymbol{a}$, $\boldsymbol{b}$, $\boldsymbol{c}$ and a list of constituent
atoms in fractional coordinates.

  * •

Cuts, are defined by a plane normal and a plane point, both given in
fractional coordinates. For example, the cut with normal [111] and plane point
(0, 0, 1) would cut along the (111) plane that passes through the point at the
grains maximum extent in the $\boldsymbol{c}$ lattice direction.

  * •

Repeat Ratio defines the ratio between unitcell repeats in the three lattice
directions. For example a repeat ratio of [1, 2, 1] would repeat the unitcell
two times in the $\boldsymbol{b}$ direction for every single repeat in the
$\boldsymbol{a}$ and $\boldsymbol{c}$ directions.

Together, Cuts and Repeat Ratio define a scalable general shape. To create
grains of definite sizes, we used an integer scaling factor, which works in
the two following ways:

  1. 1.

Multiplies the repeat ratio to get the unitcell repeats in each direction. For
example if the repeat ratio is [1, 2, 1] and you have a scaling factor of 10,
the final repetitions in each direction are [10, 20, 10].

  2. 2.

Scales the plane point of each cut. For example, if a cut is defined by the
plane normal [111] and the plane point (0, 0, 1), having a final number of
unitcell repeats equal to: [10, 20, 10], scales the plane point to (0, 0, 10).

With size factor, we have all the required information to build a grain. We
create the grain by using the repeat ratio to turn the unitcell into a
supercell, and cutting the supercell along the defined planes.

#### Composition Matching

To compare grain preference using Boltzmann factors they must have the exact
same chemical composition.

To achieve this, we used a composition matching algorithm. The process starts
with defining two or more grain shapes, then selecting a grain size target,
measured in atoms. Each grain shape is scaled so that it’s close to, but
above, the target number of atoms. We call grains in this state "as built".

Each as built grain’s element ratio is used to define a best fit composition.
Using FePt L10 as an example, grain ratios of 49:51, 49.5:50.5, and 51:49
(Fe:Pt), give you an average of 49.83:50.17, which is used to decide the
number of each atom type in the grains. Continuing the example, for grains
with 10,000 atoms total, the best fit composition would be
0.4983$\times$10,000 = 4983 iron atoms, and 0.5017$\times$10,000 = 5017
platinum atoms.

To match the compositions, atoms must be removed from the surface of the
grains; defining a best fit composition ensures that each grain’s surface will
always have the desired atoms for removal.

Once the best fit composition has been defined, atoms are progressively
removed from the surface of the as built grains, until they match the desired
composition. Atoms are preferentially removed from the least energetically
favourable positions first, starting with corner atoms, then edge atoms, and
finally general surface atoms. This produces physically realistic grains.

An example of the as built, and the compositionally matched final grain can be
seen in Figure 3.

As the figure shows, the as built grain has clean surfaces with defined edges,
in contrast the compositionally matched grain has rounded edges, although it
maintains the same underlying morphology.

#### FePt Grain Shapes

(a)

(b)

(c)

Figure 6: 2D example of how point placement affects grain morphology, when
cuts are made along the {11} plane set. The blue cube represents the original
cubic supercell and the coloured diamonds represent the area left untouched by
the {11} cuts. The final grain shape is given by the area of the blue cube
that is covered by the coloured diamond. The points which define the placement
of cuts are noted in each diagram. The figure shows: a) The 2D representation
of the Octahedron shape, b) The 2D representation of the Truncated Octahedron
Minor shape, and c) The 2D representation of the Truncated Octahedron Major
shape.

To choose our grain morphologies, we calculated the surface energies of L10’s
low index planes to see which should dominate our grains’ surfaces. Our
calculated surface energies can be see in Table 1 alongside surface
calculations taken from a paper by Kim et al.[22].

Our calculations showed that {111} surfaces have the lowest energy, due to a
greater number of interplanar bonds (6 vs 4).

Using this as a basis, all our grain shapes used cuts along the {111} planes
to expose as much of this preferential surface as possible; {100} and {001}
planes, which naturally accompany {111} plane cuts, were left to compose the
remaining surface. As noted in the results, we looked at five types of grain
shape, three with {111} dominated surfaces: Octahedron, Truncated Octahedron
Minor, and Truncated Octahedron Major; and two with energetically unpreferable
surfaces: Sphere and Cuboid. The five grain types can be seen in Figures 1 and
2.

For each grain, we used an FePt L10 unitcell with the following parameters:
$a=b=3.83$Å, $c=3.711$Å, and $\alpha=\beta=\gamma=90\degree$[26], where $a$,
$b$, $c$ are the lattice parameters, and $\alpha$, $\beta$, $\gamma$ are the
angles between them. Each set of {111} plane cuts, used to create the three
none extreme grain shapes, used two plane points, $P_{1}$ and $P_{2}$,
selected along the positive and negative $a$ direction. The points were chosen
to obey the property: $|\boldsymbol{P_{1}}|=|\boldsymbol{P_{2}}|$, assuring
the as built grains had a symmetrical structure.

The shape of each grain was decided by changing the distance of each point
from the centre of the supercell. Figure 6 demonstrates how point placement
affects the final shape of 2D as built grains.

As can be seen in Figure 6, the area of the cube covered by the diamond
creates a similar grain shape to our 3D cases projected onto a 2D plane. The
edges of the diamond passing through the plane points can be thought of as the
{11} planes associated with that particular point. In 2D there are only two
planes associated with each point, but for the 3D case each point has four
{111} planes associated with it.

### Simulation Methodology

We used MEAM potentials within the LAMMPS molecular dynamics package to
perform energy minimisation calculations.

#### MEAM Potentials

The constants for our MEAM potentials are given in Table’s 2 and 3.

Parameter | Fe | Pt
---|---|---
Ec | 4.29 | 5.77
Re | 2.867 | 3.92
B | 1.08 | 1.80
A | 0.56 | 0.90
$\beta^{(0)}$ | 4.15 | 4.92
$\beta^{(1)}$ | 1.0 | 2.2
$\beta^{(2)}$ | 1.0 | 6.0
$\beta^{(3)}$ | 1.0 | 2.2
$t^{(0)}$ | 1.0 | 1.0
$t^{(1)}$ | 2.6 | 3.94
$t^{(2)}$ | 1.8 | -2.20
$t^{(3)}$ | -7.2 | 3.84
Table 2: MEAM potential parameters for iron and platinum. Ec is the reference structure cohesive energy, Re is the reference structure equilibrium bond distance, B is the Bulk modulus, A is a model parameter that scales the screening functions, $\beta^{(l)}$ $l$=1-3 are scaling parameters controlling the form of the original EAM partial electron density functions: $\rho_{i}^{a(l)}$, and $t^{(0)}$ which scales the contribution of each of the MEAM partial electron densities: $\rho_{i}^{(l)}$ $l$=1-3. Parameter | Value
---|---
C_min(Fe-Fe-Fe) | 0.36
C_max(Fe-Fe-Fe) | 2.80
C_min(Pt-Pt-Pt) | 1.53
C_max(Pt-Pt-Pt) | 2.80
C_min(Pt-Fe-Fe) | 0.36
C_max(Pt-Fe-Fe) | 2.80
C_min(Fe-Pt-Pt) | 1.53
C_max(Fe-Pt-Pt) | 2.80
C_min(Fe-Fe-Pt) | 0.844
C_max(Fe-Fe-Pt) | 2.80
C_min(Pt-Fe-Pt) | 0.844
C_max(Pt-Fe-Pt) | 2.80
E_c(Fe, Pt) | 5.86
R_e(Fe, Pt) | 2.781
Table 3: The MEAM potential constants governing interactions between iron and
platinum. The C_min and C_max constants are the maximum and minimum values for
the screening functions when atoms are screened by an intervening atom, for
example C_min(Pt-Fe-Pt) is the maximum screening function due to an iron atom
screening the interaction of two platinum interactions. E_c(Fe, Pt) is the
energy of the iron - platinum reference structure, and R_e(Fe, Pt) is the
equilibrium bond distance of the iron - platinum reference structure.

The potentials were taken from a paper by Kim et al.[22] and Table 1 shows a
comparison between our calculated surface energies and Kim et al.’s surface
energies. The comparison shows that our calculations are in good agreement
with Kim et al. in the {111}, {101} and {110} planes, but in poor agreement
for the {100} and {001} planes. The exact reason for this isn’t clear,
although it’s possible the discrepancy is due to different surface energy
calculation methods. However, as our work is primarily intended to demonstrate
a methodology and the trend of the potentials is roughly the same, we deemed
the potentials accurate enough for this work.

#### Simulation Methodology

Before performing energy minimisation, each grain was equilibrated in an NPT
ensemble to avoid convergence to a local minimum. The NPT ensemble is set at
1K with a pressure of 0Pa and evolved the system over 15,000 time steps. The
simulation’s time steps were 0.001ps, making this phase 15ps long.
Minimisation used the standard conjugate gradient algorithm in LAMMPS and
occured over a minimum of 10,000 time steps to ensure the maximum level of
covergence. Generally, convergence was reached much sooner.

Figure 7: Left) Atomistic visualisation of an FePt L10 Truncated Octahedron
Minor grain, Right) Surface of the grain extracted by PyVista, arrows out of
the surfaces indicating the plane normals.

### Probability Calculations

To calculate the probability of each grain shape we used their minimised
energy to produce their Boltzmann factors via the following equation:

$p_{i}\propto e^{\frac{-\epsilon_{i}}{k_{B}T}}$ (1)

where $p_{i}$ is the probability of grain $i$, $\epsilon_{i}$ is the minimised
energy of grain $i$, $k_{B}$ is the Boltzmann constant, and $T$ is the
temperature. Normalising each grain’s Boltzmann factor by the minimum energy
grain’s Boltzmann factor gave us their relative probabilities. Plotting this
gave us Figure 4.

As they are compositionally identical, the grains are rearrangements of the
same system. Therefore, taking the ratio of their Boltzmann factors is the
same as taking the ratio of their probabilities, and this method allowed us to
make predictions about the most probable structure.

### Surface Energy Calculations

To investigate what role surface energy plays in shape probability
probability, we developed an algorithm that extracts the surface from the
grain shapes.

We used PyVista[27], which is a wrapper around VTK[28], to extract the
surfaces using Delaunay triangulation[29]. Used on each grain, this algorithm
gave us a collection of triangles (2D simplexes) that made up the grain’s
surface. We used the triangle’s points to calculate their plane normals and
area. Grouping the triangles by their plane normals allowed us to calculate
the surface area associated with each plane, e.g. {111}, {001}, etc.

Finally, to get the each grain’s surface energy, we multiplied their surface
areas by our calculated surface energies per unit area shown in Table 1. This
process is shown visually in Figure 7 and the code can be found in the modules
contained in this folder, in the Grain Modeller Python package.

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## Author contributions statement

G.H. directed the project, C.S. developed the Python Package, C.S. wrote the
Python package, C.S. and G.H. conceived the models, C.S. built the models,
C.S. ran the simulations, C.S. and G.H. analysed the results, and C.S. and
G.H. wrote the paper. All authors reviewed the manuscript.

## Acknowledgements

Thank you to Toyota for sponsoring this work.

## Additional information

The authors declare no competing interests.


# raceBERT – A Transformer-based Model for Predicting Race and Ethnicity from
Names111https://github.com/parasurama/raceBERT

Prasanna Parasurama
New York University<EMAIL_ADDRESS>
###### Abstract

This paper presents raceBERT – a transformer-based model for predicting race
and ethnicity from character sequences in names, and an accompanying python
package. Using a transformer-based model trained on a U.S. Florida voter
registration dataset222I sincerely thank Gaurav Sood and Suriyan Laohaprapanon
for sharing their data, the model predicts the likelihood of a name belonging
to 5 U.S. census race categories (White, Black, Hispanic, Asian & Pacific
Islander, American Indian & Alaskan Native). I build on [9] by replacing their
LSTM model with transformer-based models (pre-trained BERT model, and a
roBERTa model trained from scratch), and compare the results. To the best of
my knowledge, raceBERT achieves state-of-the-art results in race prediction
using names, with an average f1-score of 0.86 – a 4.1% improvement over the
previous state-of-the-art, and improvements between 15-17% for non-white
names.

##  1 Introduction

Researchers studying racial disparities often do not have self-reported
demographic data readily available, and must rely on proxies such as name and
location to predict race (e.g. [13, 4]). Similary, researchers studying racial
discrimination are often interested in racial signal encoded in names [2, 6].
In hiring discrimination studies, for example, the recruiter’s perception of a
candidate’s race (as proxied by name) becomes important, in which case it’s
useful to estimate the likelihood of name belonging to a particular race [8].
Over the years, new methods and models have been proposed to incrementally
improve the accuracy of race prediction models [4, 5, 1, 9, 12]. This paper
contributes to this line of research by presenting a transformer-based race
and ethnicity prediction model, which, to the best of my knowledge, achieves
state-of-the-art results in predictive accuracy.

##  2 Data

The race prediction model uses the U.S. Florida voter registration
dataset333https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/UBIG3F
collected by [9]. In total, there are 13,089,545 names across 5 race
categories. Table 1 reports the counts and label for each race category. Note
that unknown, other, and multiracial categories are dropped, because there are
too few examples.

Race | Label | Count
---|---|---
Non-hispanic White | nh_white | 8,757,268
Hispanic | hispanic | 2,179,106
Non-hispanic Black | nh_black | 1,853,690
Asian & Pacific Islander | api | 253,808
American Indian & Alaskan Native | aian | 45,673
Table 1: Florida voter registration data race label and counts

##  3 Models

The main point of departure from [9] is that I replace their LSTM classifier
with transformer-based models.

###  3.1 Fine-tuning a pre-trained BERT model

As a baseline, I start with a pretrained BERT444https://huggingface.co/bert-
base-uncased model for sequence classification [11, 3]. Although BERT
classification typically operates at the sentence level with sequences of
tokens, it can also be used at the token level with the sequences of word
pieces (i.e. sequences of characters). First, in the preprocessing step, the
name is lowercased, and the first name and the last name are concatenated by
an underscore. For example George Smith becomes george_smith. Then the
normalized name gets tokenized into words or word pieces depending on whether
the name is in the model’s vocabulary. For example, george_smith gets
tokenizen into [[CLS], george, _, smith, [SEP]]. Here, george and smith become
distinct tokens because both names are part of the vocabulary. If a name is
not in the vocabulary, it gets tokenized into word pieces. For example,
satoshi and nakamoto are not in the vocabulary, therefore satoshi_nakamoto
gets tokenized into [[CLS], sato, ##shi, _, nak, ##amo, ##to, [SEP]]

For the training step, following [10], I use the following hyperparameters:
N_EPOCHS=4, BATCH_SIZE_PER_GPU=128, LEARNING_RATE=2e-5, WEIGHT_DECAY=2e-5. See
the wandb project page for a complete list of hyperparameters and configs.

Table 2 reports the performance of the pre-trained model on a hold-out test
set, and Table 3 compares raceBERT’s performance to ethnicolr’s performance
(Note that ethnicolr does not have aian as a category). raceBERT achieves
better performance across all race categories compared to ethnicolr. On
average, there is a 4.4% improvement in the f1-score, with the highest
performance improvement coming from non-white names, with improvements as much
as 18% for black names.

Group | Precision | Recall | f1-score
---|---|---|---
nh_white | 0.89 | 0.94 | 0.91
nh_black | 0.80 | 0.58 | 0.67
hispanic | 0.85 | 0.88 | 0.87
api | 0.79 | 0.66 | 0.72
aian | 0.60 | 0.01 | 0.02
Average | 0.87 | 0.87 | 0.86
Table 2: raceBERT hold-out performence metrics (pre-trained language model) Group | Precision | Recall | f1-score |  ethnicolr f1-score | % Improvement
---|---|---|---|---|---
nh_white | 0.89 | 0.94 | 0.91 | 0.90 | 1.51
nh_black | 0.80 | 0.58 | 0.67 | 0.55 | 18.15
hispanic | 0.85 | 0.88 | 0.87 | 0.75 | 13.53
api | 0.80 | 0.66 | 0.72 | 0.60 | 16.89
Average | 0.87 | 0.87 | 0.87 | 0.83 | 4.38
Table 3: raceBERT hold-out performence improvement (pre-trained language
model)

I repeat the training process with the Wikipedia ethnicity dataset [1, 9] and
report the results in Appendix A. As with the race model, raceBERT achieves an
overall 5% improvement over ethnicolr.

###  3.2 Training a model from scratch

Although the pre-trained BERT model achieves significant performance
improvements over ethnicolr, a few limitations remain. First, with a
vocabulary size of 30,000, the model is needlessly large for the task at hand.
Most of the tokens in the vocabulary will never be used. Second, many commonly
occurring names are in the vocabulary, which raises the question of whether
the model is simply memorizing names rather than learning from character
sequences of names, in which case the generalizability of the model will
suffer. To overcome these issues, I train a roBERTa model from scratch with a
much smaller vocabulary size of 500 [7].

In the preprocessing step, the first name is lowercased, and the last name is
uppercased, which are then concatenated by a space. For example George Smith
becomes george SMITH. In theory, the mixed lower/upper case will make it
easier for the model to discriminate between first and last names. Using the
transformed names, I train a Byte Pair Encoding tokenizer with a max
vocabulary size of 500 to learn the most commonly occurring character
sequences, which is then used to tokenize all names. For example, george SMITH
gets tokenized into [[CLS], ge, or, ge, SMITH, [SEP]]. Likewise, satoshi
NAKAMOTO gets tokenized into [[CLS], sa, t, o, sh, i, N, A, K, AM, O, T, O,
[SEP]].

Next, I train a masked language model to learn the character sequences using
the roBERTa masked language architecture (ATTENTION_HEADS=12,
HIDDEN_LAYERS=6). Using the weights from the language model, I initialize a
roBERTa sequence classification model and train using the following
hyperparameters: N_EPOCHS=4, BATCH_SIZE_PER_GPU=128. LEARNING_RATE=2e-5,
WEIGHT_DECAY=2e-5. All other configs can be found at the wandb project page.

Table 4 reports the performance of the model on a hold-out test set, and Table
5 reports the performance improvement compared to ethnicolr. The model’s
performance metrics are almost the same as the pre-trained BERT model, but
with a much smaller vocabulary and (theoretically) greater generalizability.
As such, I use this as the default model for the python package.

Group | Precision | Recall | f1-score
---|---|---|---
nh_white | 0.89 | 0.94 | 0.91
nh_black | 0.79 | 0.57 | 0.66
hispanic | 0.85 | 0.88 | 0.86
api | 0.78 | 0.64 | 0.70
aian | 0.53 | 0.01 | 0.01
Average | 0.86 | 0.87 | 0.86
Table 4: raceBERT hold-out performence metrics metric | precision | recall | f1 | ethnicolr f1 | % improvement
---|---|---|---|---|---
group |  |  |  |  |
nh_white | 0.89 | 0.94 | 0.91 | 0.90 | 1.37
nh_black | 0.80 | 0.57 | 0.67 | 0.55 | 17.39
hispanic | 0.85 | 0.88 | 0.87 | 0.75 | 13.32
api | 0.79 | 0.64 | 0.71 | 0.60 | 15.11
Average | 0.87 | 0.87 | 0.87 | 0.83 | 4.11
Table 5: raceBERT hold-out performence improvements

###  3.3 Code, Configs, and Resources

All of the training code, as well as the raceBERT python package code, is on
Github555https://github.com/parasurama/raceBERT. The trained models are
uploaded to the huggingface hub666https://huggingface.co/pparasurama/raceBERT.
All training configurations, hyperparameters, and learning curves can be found
at the wandb project page777https://wandb.ai/parasu/raceBERT-public.

##  4 Conclusion & Limitations

This paper presents a new transformer-based model for predicting race from
names and demonstrates the performance improvements over existing state-of-
the-art models. One limitation of this model is that it’s trained on the
Florida voter registration dataset, which is not necessarily representative of
the U.S. population. As such, the accuracy scores may vary when used on
datasets from U.S. regions with different demographic distributions. Another
limitation – one that’s shared by all race prediction models – is that the
model is not 100% accurate. While this is admissable when studying racial
disparities in the aggregate, or when estimating the likelihood of a name
belonging to a particular race, it’s not advisable to use this model in
applications where it is important to know individual race.

## References

  * [1] Anurag Ambekar et al.  “Name-Ethnicity Classification from Open Sources”  In _Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD ’09_ Paris, France: ACM Press, 2009, pp. 49  DOI: 10.1145/1557019.1557032
  * [2] Marianne Bertrand and Sendhil Mullainathan  “Are Emily and Greg More Employable Than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination”  In _THE AMERICAN ECONOMIC REVIEW_ 94.4, 2004, pp. 79
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  * [4] Kevin Fiscella and Allen M. Fremont  “Use of Geocoding and Surname Analysis to Estimate Race and Ethnicity”  In _Health Services Research_ 41.4p1, 2006, pp. 1482–1500  DOI: 10.1111/j.1475-6773.2006.00551.x
  * [5] Kosuke Imai and Kabir Khanna  “Improving Ecological Inference by Predicting Individual Ethnicity from Voter Registration Records”  In _Political Analysis_ 24.2 Cambridge University Press, 2016, pp. 263–272  DOI: 10.1093/pan/mpw001
  * [6] Sonia K. Kang, Katherine A. DeCelles, András Tilcsik and Sora Jun  “Whitened Résumés: Race and Self-Presentation in the Labor Market”  In _Administrative Science Quarterly_ 61.3 SAGE Publications Inc, 2016, pp. 469–502  DOI: 10.1177/0001839216639577
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  * [8] Prasanna Parasurama, Anindya Ghose and Panos Ipeirotis  “Who Benefits from Organizational Diversity Efforts? Evidence from Silicon Valley Firms”  In _Academy of Management Proceedings_ 2020.1 Academy of Management, 2020, pp. 22036  DOI: 10.5465/AMBPP.2020.22036abstract
  * [9] Gaurav Sood and Suriyan Laohaprapanon  “Predicting Race and Ethnicity From the Sequence of Characters in a Name”  In _arXiv:1805.02109 [stat]_ , 2018  arXiv:1805.02109 [stat]
  * [10] Chi Sun, Xipeng Qiu, Yige Xu and Xuanjing Huang  “How to Fine-Tune BERT for Text Classification?”  In _Chinese Computational Linguistics_ , Lecture Notes in Computer Science  Cham: Springer International Publishing, 2019, pp. 194–206  DOI: 10.1007/978-3-030-32381-3˙16
  * [11] Ashish Vaswani et al.  “Attention Is All You Need”  In _Advances in Neural Information Processing Systems_ 30 Curran Associates, Inc., 2017, pp. 5998–6008
  * [12] Fangzhou Xie  “Rethnicity: Predicting Ethnicity from Names”  In _arXiv:2109.09228 [cs]_ , 2021  arXiv:2109.09228 [cs]
  * [13] Yan Zhang  “Assessing Fair Lending Risks Using Race/Ethnicity Proxies”  In _Management Science_ 64.1, 2018, pp. 178–197  DOI: 10.1287/mnsc.2016.2579

## Appendix

##  Appendix A Wikipedia Ethnicity Dataset

Ethnicity | Count
---|---
GreaterEuropean,British | 43,579
GreaterEuropean,WestEuropean,French | 14,076
GreaterEuropean,WestEuropean,Italian | 13,633
GreaterEuropean,WestEuropean,Hispanic | 11,365
GreaterEuropean,Jewish | 11,224
GreaterEuropean,EastEuropean | 9,318
Asian,IndianSubContinent | 9,063
Asian,GreaterEastAsian,Japanese | 8,425
GreaterAfrican,Muslim | 6,854
Asian,GreaterEastAsian,EastAsian | 6,514
GreaterEuropean,WestEuropean,Nordic | 5,242
GreaterEuropean,WestEuropean,Germanic | 4,634
GreaterAfrican,Africans | 4,348
Table 6: Wikipedia dataset ethnicity categories and counts

Group | Precision | Recall | f1-score |  ethnicolr f1-score | %Improvement
---|---|---|---|---|---
Asian,GreaterEastAsian,EastAsian | 0.87 | 0.85 | 0.86 | 0.82 | 4.65
Asian,GreaterEastAsian,Japanese | 0.92 | 0.88 | 0.90 | 0.90 | 0.00
Asian,IndianSubContinent | 0.82 | 0.79 | 0.81 | 0.77 | 4.94
GreaterAfrican,Africans | 0.65 | 0.52 | 0.58 | 0.49 | 15.52
GreaterAfrican,Muslim | 0.66 | 0.76 | 0.71 | 0.67 | 5.63
GreaterEuropean,British | 0.82 | 0.87 | 0.85 | 0.82 | 3.53
GreaterEuropean,EastEuropean | 0.81 | 0.78 | 0.80 | 0.76 | 5.00
GreaterEuropean,Jewish | 0.57 | 0.53 | 0.55 | 0.47 | 14.55
GreaterEuropean,WestEuropean,French | 0.71 | 0.68 | 0.69 | 0.64 | 7.25
GreaterEuropean,WestEuropean,Germanic | 0.54 | 0.56 | 0.55 | 0.46 | 16.36
GreaterEuropean,WestEuropean,Hispanic | 0.74 | 0.73 | 0.73 | 0.70 | 4.11
GreaterEuropean,WestEuropean,Italian | 0.77 | 0.79 | 0.78 | 0.75 | 3.85
GreaterEuropean,WestEuropean,Nordic | 0.82 | 0.74 | 0.78 | 0.68 | 12.82
Average | 0.77 | 0.77 | 0.77 | 0.73 | 5.19

Table 7: raceBERT hold-out performance on Wikipedia ethnicity dataset


# Some applications of the Menshov–Rademacher theorem

Safari Mukeru
Department of Decision Sciences
University of South Africa, P. O. Box 392, Pretoria, 0003. South Africa
e-mail<EMAIL_ADDRESS>
###### Abstract

Given a sequence $(X_{n})$ of real or complex random variables and a sequence
of numbers $(a_{n})$, an interesting problem is to determine the conditions
under which the series $\sum_{n=1}^{\infty}a_{n}X_{n}$ is almost surely
convergent. This paper extends the classical Menshov–Rademacher theorem on the
convergence of orthogonal series to general series of dependent random
variables and derives interesting sufficient conditions for the almost
everywhere convergence of trigonometric series with respect to singular
measures whose Fourier transform decays to 0 at infinity with positive rate.

Key words: Dependent random series, Menshov–Rademacher theorem, trigonometric
series.

##  1 Introduction

A classical fundamental result obtained independently by Menshov and
Rademacher in the 1920’s states that if the condition
$\sum_{n=1}^{\infty}|a_{n}|^{2}\log_{2}^{2}(n+1)<\infty$ is satisfied for a
given sequence of real or complex numbers $(a_{n})$, then for any sequence of
orthonormal sequence $(\varphi_{n})$ of $L^{2}$ functions on the unit circle
$\mathbb{T}$, the series $\sum_{k=1}^{\infty}a_{n}\varphi(x)$ converges almost
everywhere on $\mathbb{T}$ (with respect to the Lebesgue measure). This
condition has been extended to the $L^{2}$ space of general measure spaces and
more recently Paszkiewicz [12, 13] and Bednorz [2] obtained a necessary and
sufficient condition on the sequence $(a_{n})$ for the convergence a.e. of
$\sum_{k=1}^{\infty}a_{n}\varphi(x)$ (the condition is expressed in terms of
the existence of a majorising measure on the set
$\\{\sum_{k=1}^{n}a_{k}^{2}:n\geq 1\\}\cup\\{0\\}$). Menshov and Rademacher
theorem was probably the first result related to random series of non-
independent random variables. The problem of convergence of random series with
dependent random variables and the problem of extending limit theorems of
probability theory related to iid variables to dependent random variables has
attracted a lot of attention since the introduction of sequences of negatively
associated random variables by Joag-Dev and Proschan [4]. However the problem
remains largely open.

The problem of dependent random variables is very wide and it is difficult to
find a general criteria that applies to all. In the literature the general
approach is to delineate certain forms of dependency and classes of random
variables and determine an appropriate condition for the a.s. convergence.
Matuła [9] proved that if $(X_{n})$ are negatively associated real random
variables with finite second moments then the condition
$\sum_{n=1}^{\infty}\mathbb{E}(|X_{n}|^{2})<\infty$ implies the a.s.
convergence of $\sum_{n=1}^{\infty}X_{n}$. This has been extended to random
vectors in a Hilbert space by Ko, Kim and Han [7].

Antonini, Kozachenko and Volodin [1] considered the case of sub-Gaussian
random variables $(X_{n})$ exhibiting certain dependence structures (negative
association, $m$-dependence and $m$-acceptability) and derived interesting
sufficient conditions for the a.s. convergence. It is shown in [11] that if
$(X_{n})$ can be expressed as linear combinations
$X_{n}=\sum_{k=1}^{n}a_{n,k}Z_{k}$ for a fixed sequence $(Z_{n})$ of i.i.d
random variables (with zero mean and unit variance) and complex numbers
$(a_{n,k})$, then the condition
$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}|a_{n+k-1,k}|^{2}\right)^{1/2}<\infty$
is sufficient for the a.s. convergence of the series
$\sum_{n=1}^{\infty}X_{n}$. (For the problem of limit theorems of sequences of
dependent random variables we refer to Naderi et al. [3] and references
therein for some recent developments.)

In this paper we consider the problem of convergence of the random series
$\sum_{n=1}^{\infty}a_{n}X_{n}$ where $(a_{n})$ is a sequence of real or
complex numbers and $(X_{n})$ a sequence of real or complex random variables
in the most general dependence case. Our approach consists of extending the
Menshov–Rademacher theorem from the classical case of orthonormal series to
general dependent random variables. We thus obtain an explicit sufficient
condition for the almost sure convergence of random series in their most
generality. This condition is also applied to study almost everywhere
convergence of trigonometric Fourier series on some subsets of the unit circle
of Lebesgue measure zero.

Our results can be summarised are as follows: If
$\sum_{m,n=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|\log_{2}(n+1)\log_{2}(m+1)<\infty$,
then the series $\sum_{n=1}^{\infty}a_{n}X_{n}$ converges a.s. From this
result, we derive the following: If the linear operator defined by the matrix
$A=(|\mathbb{E}(X_{n}\overline{X_{m}})|)$ in $\ell^{2}$ is bounded, then
Menshov–Rademacher condition
$\sum_{n=1}^{\infty}|a_{n}|^{2}\log_{2}^{2}(n+1)<\infty$ for the a.s.
convergence of orthonormal series is also sufficient for the a.s. convergence
of the series $\sum_{n=1}^{\infty}a_{n}X_{n}.$ The same is true if $(a_{n})$
are real numbers and $(X_{n})$ are real random variables such that
$\mathbb{E}(X_{n}X_{m})\leq 0$ for all $m\neq n$. If the matrix
$(n^{-b}m^{-b}A_{n,m})$ is bounded for some $b\geq 0$, then the condition
$\sum_{n=1}^{\infty}|a_{n}|^{2}n^{2b}\log_{2}^{2}(n+1)<\infty$ is sufficient
for the a.s. convergence of $\sum_{n=1}^{\infty}a_{n}X_{n}$. Finally if $\mu$
is a Borel probability measure on the unit circle $\mathbb{T}$ such that its
Fourier transform satisfies $|\hat{\mu}(n)|\leq K|n|^{-a}$ for some $a\geq 0$
and there exists $b>(1-a)/2$ such that
$\sum_{n\in\mathbb{Z}}|a_{n}|^{2}|n|^{2b}\log_{2}^{2}(n+1)<\infty$, then the
Fourier trigonometric series $\sum_{n\in\mathbb{Z}}a_{n}\exp(2\pi int)$
converges $\mu$-almost everywhere on $\mathbb{T}$. This implies that the
Fourier series of a well-behaved $L^{2}$ function $f$ on $\mathbb{T}$ in the
sense that $\sum_{n=1}^{\infty}|\hat{f}(n)|^{2}n^{2b}\log_{2}^{2}(n+1)<\infty$
for some number $0\leq b\leq 1/2$ cannot diverge everywhere on a subset of
Fourier dimension $>1-2b$. In particular if a function belongs to the Sobolev
space $H^{p}(\mathbb{T})$ for some $0\leq p\leq 1/2$, then its Fourier series
converges $\mu$-almost everywhere on every subset $E$ of $\mathbb{T}$ of
Fourier dimension $>1-2p$ where $\mu$ is a Borel measure supported by $E$ such
that $|\mu(n)|=o(|n|^{-(1-2p)}).$

The paper concludes with a remark concerning the particular case of Gaussian
random variables were we prove (using the classical Sudakov–Fernique
inequality) that the condition
$\sum_{m,n=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|<\infty$
is enough for the a.s. convergence of $\sum_{n=1}^{\infty}a_{n}X_{n}$.

##  2 An extension of Menshov–Rademacher theorem

For a sequence $(a_{n})$ of numbers the classical Menshov–Rademacher theorem
says that if the condition

$\displaystyle\sum_{n=1}^{\infty}|a_{n}|^{2}\log_{2}^{2}(n+1)<\infty,$ (1)

then the series $\sum_{n=1}^{\infty}a_{n}\varphi(x)$ converges almost
everywhere for any sequence $(\varphi_{n})$ of orthonormal functions in
$L^{2}(\mathbb{T})$. (Here $\mathbb{T}$ is the unit circle
$\mathbb{R}/\mathbb{Z}$.) In this paper we relax the condition of
orthogonality and extend Menshov–Rademacher theorem to series
$\sum_{n=1}^{\infty}a_{n}\varphi(x)$ in that general case. In general we
consider random variables in the $L^{2}(\Omega)$ space of a certain
probability space. The usual inner product in $L^{2}(\Omega)$ is $\langle
X,Y\rangle=\mathbb{E}(X\overline{Y}).$

###### Theorem 1

For any sequence of real or complex numbers $(a_{n})$ and a sequence $(X_{n})$
of real or complex random variables such that $\mathbb{E}(|X_{n}|^{2})=1$ for
all $n$, if

$L:=\sum_{n,m=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|\log_{2}(n+1)\log_{2}(m+1)<\infty,$

then the series $\sum_{n=1}^{\infty}a_{n}X_{n}$ converges almost surely.
Moreover,

$\mathbb{E}\left(\sup_{n\in\mathbb{N}}|a_{1}X_{1}+a_{2}X_{2}+\ldots+a_{n}X_{n}|^{2}\right)\leq
8L.$

The proof is based on the classical proof of the Menshov–Rademacher theorem as
given in Kashin and Saakyan [6, p 251]. We have also used a proof given by
Mikhailets and Murach [10]. The following lemma is also an extension of the
classical Menshov - Rademacher lemma.

###### Lemma 1

With the hypothesis of the theorem we have the following inequality: For any
integer $N>0$, let

$S_{N}^{*}=\max_{1\leq j\leq N}|\sum_{n=1}^{j}a_{n}X_{n}|.$

Then

$\mathbb{E}\left(S_{N}^{*}\right)^{2}\leq(2+\log_{2}N)^{2}\sum_{n,m=1}^{N}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

  * Proof

The starting point is to assume that $N=2^{r}$ for some integer $r\geq 1$ and
show that

$\mathbb{E}\left(S_{N}^{*}\right)^{2}\leq(1+\log_{2}N)^{2}\sum_{n,m=1}^{N}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

The case where $2^{r-1}<N\leq 2^{r}$ is reduced to the previous one by just
taking $a_{n}=0$ for all $n$ such that $N<n\leq 2^{r}$. Then this inequality
holds for $r-1<\log_{n}N$, that is,

$\mathbb{E}\left(S_{N}^{*}\right)^{2}\leq(2+\log_{2}N)^{2}\sum_{n,m=1}^{N}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

Now for $N=2^{r}$, consider for each number $j\in\\{1,2,\ldots,2^{r}\\}$ its
dyadic representation

$j=\sum_{k=0}^{r}\xi_{k}(j)2^{r-k},\,\,\mbox{ with }\xi_{k}(j)=0\mbox{ or
}1,\mbox{ for all }k.$

Moreover for each such $j$, decompose the interval $[0,j]$ in $\mathbb{N}$
into subintervals

$[1,j]=\bigcup_{k:\xi_{k}(j)\neq 0}I_{k}$

where

$I_{k}:=I_{k}(j)=\left\\{n\in\mathbb{N}:\sum_{s=0}^{k-1}\xi_{s}(j)2^{r-s}<n\leq\sum_{s=0}^{k}\xi_{s}(j)2^{r-s}\right\\}.$

Now write

$\sum_{n=1}^{j}a_{n}X_{n}=\sum_{k:\xi_{k}(j)\neq 0}\sum_{n\in
I_{k}(j)}a_{n}X_{n}.$

Then

$\displaystyle|\sum_{n=1}^{j}a_{n}X_{n}|$ $\displaystyle=$
$\displaystyle|\sum_{k:\xi_{k}(j)\neq 0}\sum_{n\in I_{k}}a_{n}X_{n}|$
$\displaystyle\leq$ $\displaystyle\sum_{k:\xi_{k}(j)\neq 0}1.|\sum_{n\in
I_{k}}a_{n}X_{n}|$ $\displaystyle\leq$
$\displaystyle\left(\sum_{k:\xi_{k}(j)\neq
0}1^{2}\right)^{1/2}.\left(\sum_{k:\xi_{k}(j)\neq 0}|\sum_{n\in
I_{k}(j)}a_{n}X_{n}|^{2}\right)^{1/2}$

Clearly

$\sum_{k:\xi_{k}(j)\neq 0}1\leq\sum_{k=0}^{r}1=(1+r)$

and for all $j$,

$\sum_{k:\xi_{k}(j)\neq 0}|\sum_{n\in
I_{k}(j)}a_{n}X_{n}|^{2}\leq\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\left|\sum_{n=p2^{r-k}+1}^{(p+1)2^{r-k}}a_{n}X_{n}\right|^{2}.$

Hence simultaneously for all $1\leq j\leq N$,

$\displaystyle|\sum_{n=1}^{j}a_{n}X_{n}|^{2}\leq(r+1)\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\left|\sum_{n=p2^{r-k}+1}^{(p+1)2^{r-k}}a_{n}X_{n}\right|^{2}.$

Now assume

$S_{N}^{*}=|\sum_{n=1}^{j}a_{n}X_{n}|,\,\,\,\,j\mbox{ random }.$

Then

$(S_{N}^{*})^{2}\leq(r+1)\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\left|\sum_{n=p2^{r-k}+1}^{(p+1)2^{r-k}}a_{n}X_{n}\right|^{2}$

and hence

$\displaystyle\mathbb{E}(S_{N}^{*})^{2}$ $\displaystyle\leq$
$\displaystyle(r+1)\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\mathbb{E}\left|\sum_{n=p2^{r-k}+1}^{(p+1)2^{r-k}}a_{n}X_{n}\right|^{2}$
$\displaystyle=$
$\displaystyle(r+1)\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\sum_{n,m=p2^{r-k}+1}^{(p+1)2^{r-k}}a_{n}\overline{a_{m}}\,\mathbb{E}(X_{n}\overline{X_{m}})$
$\displaystyle\leq$
$\displaystyle(r+1)\sum_{k=0}^{r}\sum_{p=0}^{2^{k}-1}\sum_{n,m=p2^{r-k}+1}^{(p+1)2^{r-k}}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|$
$\displaystyle\leq$
$\displaystyle(r+1)\sum_{k=0}^{r}\sum_{n,m=0}^{2^{r}}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|$
$\displaystyle\leq$
$\displaystyle(r+1)^{2}\sum_{n,m=1}^{N}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

Therefore,

$\displaystyle\mathbb{E}(S_{N}^{*})^{2}$ $\displaystyle\leq$
$\displaystyle(1+\log_{2}N)^{2}\sum_{n,m=1}^{N}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

##  3 Proof of Theorem 1

The first step is to consider the sequence of partial sums of the form

$S_{2^{k}}=\sum_{n=1}^{2^{k}}a_{n}X_{n},\,\,\,\mbox{ for }k=1,2,3,\ldots$

and show that $(S_{2^{k}})$ converges almost surely and write

$S^{*}=\sup_{0\leq k<\infty}|S_{2^{k}}|.$

Set

$\chi_{k}=\sum_{n=2^{k}}^{2^{k+1}-1}a_{n}X_{n},k=0,1,2,3,\ldots$

and then

$\mathbb{E}(|\chi_{k}|^{2})\leq\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

Then,

$\displaystyle\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|^{2})(k+1)^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}(k+1)^{2}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|(1+\log_{2}n)((1+\log_{2}m)$

because for $k$ fixed and $n\geq 2^{k}$, then $(1+k)\leq(1+\log_{2}n)$ and
similarly, $(1+k)\leq(1+\log_{2}m)$. Moreover since clearly
$(1+\log_{2}n)\leq\sqrt{2}\log_{2}(n+1)$, it follows that

$\displaystyle\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|^{2})(k+1)^{2}\leq
2\sum_{k=0}^{\infty}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|\log_{2}(n+1)\log_{2}(m+1).$

Hence

$\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|^{2})(k+1)^{2}\leq 2L<\infty.$

Moreover by the Cauchy-Schwarz inequality,

$\displaystyle\sum_{k=0}^{\infty}(\mathbb{E}(|\chi_{k}|^{2}))^{1/2}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}(\mathbb{E}(|\chi_{k}|^{2}))^{1/2}(k+1)(k+1)^{-1}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|^{2})(k+1)^{2}\right)^{1/2}\left(\sum_{k=0}^{\infty}(k+1)^{-2}\right)^{1/2}\leq
2\sqrt{L}.$

This implies

$\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|)\leq\sum_{k=0}^{\infty}(\mathbb{E}(|\chi_{k}|^{2}))^{1/2}\leq
2\sqrt{L}$

and hence Beppo Levi theorem implies that
$\sum_{k=0}^{\infty}|\chi_{k}|<\infty$ almost surely. Hence the sequence
$(S_{2^{k}})$ is a Cauchy sequence and therefore it converges almost surely.
Moreover,

$\mathbb{E}((S^{*})^{2})\leq\sum_{k=0}^{\infty}\mathbb{E}(|\chi_{k}|^{2})\leq
4L.$

Let us now consider

$S^{\circ}_{k}=\max_{2^{k}\leq
j<2^{k+1}}|\sum_{n=2^{k}}^{j}a_{n}X_{n}|,\,\,k=1,2,3,\ldots\,\,\,\mbox{ and
}S^{\circ}=\sup_{1\leq k<\infty}S^{\circ}_{k}.$

By the lemma,

$\mathbb{E}((S^{\circ}_{k})^{{}^{2}})\leq(2+\log_{2}(j-2^{k}+1))^{2}\sum_{n,m=2^{k}}^{j}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|.$

Then

$\displaystyle\sum_{k=0}^{\infty}\mathbb{E}((S^{\circ}_{k})^{{}^{2}})$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}(2+\log_{2}2^{k})^{2}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}(2+\log_{2}2^{k})^{2}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{n,m=2^{k}}^{2^{k+1}-1}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|(2+\log_{2}n)(2+\log_{2}m)$
$\displaystyle\leq$
$\displaystyle\sum_{n,m=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|(2+\log_{2}n)(2+\log_{2}m)$
$\displaystyle\leq$ $\displaystyle 4L<\infty.$

As previously Beppo Levy theorem implies that

$\displaystyle\mathbb{E}\left(\sum_{k=0}^{\infty}(S^{\circ}_{k})^{2}\right)=\sum_{k=0}^{\infty}(\mathbb{E}(S^{\circ}_{k})^{2})\leq
4L<\infty.$

Therefore $\lim_{k\to\infty}S^{\circ}_{k}=0$ almost surely. This together with
the convergence of the sequence $(S_{2^{k}})$ implies that the series
$\sum_{n=1}^{\infty}a_{n}X_{n}$ converges almost surely. Finally, since

$\sup_{m\in\mathbb{N}}|a_{n}X_{1}+a_{2}X_{2}+\ldots+a_{n}X_{n}|^{2}\leq(S^{*})^{2}+(S^{\circ})^{2}\mbox{
and }(S^{\circ})^{2}\leq\sum_{k=0}^{\infty}(S^{\circ}_{k})^{2},$

it follows that

$\mathbb{E}\left(\sup_{m\in\mathbb{N}}|a_{n}X_{1}+a_{2}X_{2}+\ldots+a_{n}X_{n}|^{2})\right)\leq(\mathbb{E}(S^{*})^{2})+(\mathbb{E}(S^{\circ})^{2})\leq
8L.$

##  4 Some important particular cases

###### Corollary 1

Let $(a_{n})$ be a sequence of real or complex numbers and $(X_{n})$ be a
sequence of real random variables such that $\mathbb{E}(X_{n}^{2})=1$ for all
$n$ and $\mathbb{E}(X_{n}X_{m})\leq 0$ for all $n\neq m$.
If $\sum_{n=1}^{\infty}|a_{n}|^{2}\log_{2}^{2}(n+1)<~{}\infty,$ then the
series $\sum_{n=1}^{\infty}a_{n}X_{n}$ converges almost surely.

  * Proof

Let $c_{n}=a_{n}\log_{2}(n+1)$ for all $n$. Since $\mathbb{E}(X_{n}X_{m})\leq
0$ for $n\neq m$, then

$\sum_{n\neq m}|c_{n}||c_{m}||\mathbb{E}(X_{n}X_{m})|=-\sum_{n\neq
m}|c_{n}||c_{m}|\mathbb{E}(X_{n}X_{m}).$

Hence

$\sum_{n,m=1}^{\infty}|c_{n}||c_{m}||\mathbb{E}(X_{n}X_{m})|=\sum_{n=1}^{\infty}c_{n}^{2}-\sum_{n\neq
m}|c_{n}||c_{m}|\mathbb{E}(X_{n}X_{m})\geq 0.$

This yields

$\sum_{n\neq
m}|c_{n}||c_{m}||\mathbb{E}(X_{n}X_{m})|\leq\sum_{n=1}^{\infty}c_{n}^{2}$

and hence

$\sum_{n,m=1}^{\infty}|c_{n}||c_{m}||\mathbb{E}(X_{n}X_{m})|\leq
2\sum_{n=1}^{\infty}c_{n}^{2}$

and the conclusion follows by the theorem.

Corollary 1 generalises previous results of Matuła [9] and Antonini,
Kozachenko and Volodin [1] on the a.s. convergence of series of linear
combinations of negatively correlated random variables. It indicates that with
respect to a.s. convergence, sequences of real random variables that are
negatively correlated behave like sequences of uncorrelated random variables.

###### Corollary 2

Let $(a_{n})$ be a sequence of real or complex numbers and $(X_{n})$ be a
sequence of real or complex random variables with covariance matrix
$\gamma(n,m))$. Assume that there exists $b\geq 0$ such that the linear
operator defined by the matrix $(\beta_{n,m})$ with

$\beta_{n,m}=|\gamma(n,m)|n^{-b}m^{-b}$

in $\ell^{2}(\mathbb{N})$ is bounded. If

$\displaystyle\sum_{n=1}^{\infty}|a_{n}|^{2}n^{2b}\log_{n}^{2}(n+1)<\infty,$
(2)

then the series $\sum_{n=1}^{\infty}a_{n}X_{n}$ converges almost surely.

In particular if $b=0$, that is, the covariance matrix $(\gamma(n,m))$ defines
a bounded operator, then we retrieve that
$\sum_{n=1}^{\infty}|a_{n}|^{2}\log_{n}^{2}(n+1)<\infty$ implies the almost
surely convergence of $\sum_{n=1}^{\infty}a_{n}X_{n}$. This is an extension of
the Menshov–Rademacher theorem from sequences of orthogonal functions to
sequences of functions $(\varphi_{n}(x))$ such that the “covariance matrix”
$((\gamma(n,m))$ given by:

$\gamma(n,m)=\int_{\mathbb{T}}\varphi_{n}(x)\overline{\varphi_{m}(x)}dx$

defines a bounded operator in $\ell^{2}.$

  * Proof

Set $c_{n}=a_{n}\log_{2}(n+1)$ so that condition (2) becomes
$\sum_{n=1}^{\infty}|c_{n}|^{2}n^{2b}<\infty$. Write

$\displaystyle\sum_{n,m=1}^{\infty}|c_{n}||c_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|=\sum_{n,m=1}^{\infty}|c_{n}||c_{m}||\gamma(n,m)|=\sum_{n,m=1}^{\infty}|c_{n}||c_{m}|n^{b}m^{b}\beta_{nm}.$

Now the boundedness of $(\beta_{n,m})$ implies that for some fixed $K>0$,

$\sum_{n,m=1}^{\infty}|c_{n}||c_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|\leq
K\sum_{n,m=1}^{\infty}|c_{n}|^{2}n^{2b}<\infty.$

That is,

$\sum_{n,m=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|\log_{2}(n+1)\log_{2}(m+1)<\infty$

and the conclusion follows.

###### Corollary 3

Assume that the covariance matrix is such that $|\gamma(n,m)|\leq K|n-m|^{-a}$
for some fixed constants $K>0$ and $a\geq 0$. Then if there exists
$b>(1-a)/2$, such that

$\sum_{n=1}^{\infty}|a_{n}|^{2}n^{2b}\log_{n}^{2}(n+1)<\infty,$

then the series $\sum_{n=1}^{\infty}a_{n}X_{n}$ converges almost surely.

It is so because one can show that the matrix $(\beta_{n,m})$ given by
$\beta_{n,m}=|n-m|^{-a}n^{-b}m^{-b}$ is bounded. (This can be obtained by
using the Schur test to the vector $x=(x_{n})$ with $x_{n}=n^{-c}$ for some
constant $c\geq 0$ such that $a+b+c>1$.)

##  5 Application to a.e. convergence of trigonometric series

In particular consider a Borel probability measure $\mu$ on $\mathbb{T}$ and
the trigonometric sequence $(e_{n})$ given by $e_{n}(x)=\exp(2\pi inx)$,
$(n\in\mathbb{Z})$. Clearly,

$\langle
e_{n},e_{m}\rangle=\int_{\mathbb{T}}e_{n}(x)\overline{e_{m}(x)}d\mu(x)=\hat{\mu}(n-m).$

Then if

$\sum_{n,m=-\infty}^{\infty}|a_{n}||a_{m}||\hat{\mu}(n-m)|\log_{2}(|n|+1)\log_{2}(|m|+1)<\infty,$

then the trigonometric series $\sum_{n=-infty}^{\infty}a_{n}\exp(2\pi inx)$
converges $\mu$-a.e. in $\mathbb{T}$. Then Corollary 3 immediately implies the
following:

###### Corollary 4

Assume that for constant $K>0$ and $a\geq 0$,

$\displaystyle|\hat{\mu}(n)|\leq K|n|^{-a}\mbox{ for all }n\neq 0.$ (3)

If there exists $b>(1-a)/2$ such that
$\sum_{n\in\mathbb{Z}}|a_{n}|^{2}|n|^{2b}\log_{2}^{2}(|n|+1)<~{}\infty,$ then
the trigonometric series $\sum_{n\in\mathbb{Z}}a_{n}\exp(2\pi int)$ converges
$\mu$-almost everywhere on $\mathbb{T}$.

Consider now a compact subset $E$ of $\mathbb{T}$ of Lebesgue measure zero
which supports a probability measure $\mu$ such that inequality (3) holds. Let
$f\in L^{2}(\mathbb{T})$ such that
$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}|n|^{2b}\log_{2}^{2}(n+1)<\infty$ for
some $b>(1-a)/2$. Then the Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\exp(2\pi nit)$ of $f$ converges $\mu$-a.e. on
$E$. In particular if $f$ is in the Sobolev space $H^{p}(\mathbb{T})$ for some
$p>(1-a)/2$ with $0\leq a\leq 1$ (that is,
$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}(1+n^{2})^{p}<\infty$), then we can
choose $b$ with $(1-a)/2<b<p$ so that
$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}|n|^{2b}\log_{2}^{2}(n+1)<\infty$ and
obtain that the Fourier series of $f$ converges $\mu$-a.e. on $E$.

We recall that for a Lebesgue measurable subset $E$ of $\mathbb{T}$, the
Fourier dimension of $E$ is the supremum of the numbers $0\leq\alpha\leq 1$
such that $E$ supports a Borel probability measure $\mu$ such that
$|\hat{\mu}(u)|^{2}=o(|u|^{-\alpha})$ for $u\to\infty$. It is well known that
the Fourier dimension is always less than or equal to the Hausdorff dimension
and the two can be different (see e.g. Kahane [5, p. 250] and Mattila [8, p.
40]).

A classical result by Ketznelson and Kahane says that if $E$ is a subset of
$\mathbb{T}$ of Lebesgue measure $0$, then there exists a continuous function
$f$ on $\mathbb{T}$ such that its Fourier series diverges everywhere on $E$.
But now our observation here is that if $E$ has positive Fourier dimension
$\alpha$, $0<\alpha\leq 1$, such function must be such that

$\sum_{n=-\infty}^{\infty}|\hat{f}(n)|^{2}|n|^{2b}\log_{2}^{2}(|n|+1)=\infty\,\,\,\mbox{
for all }b>(1-\alpha)/2.$

In other words the Fourier series of a well-behaved $L^{2}$ function $f$ in
the sense that
$\sum_{n=1}^{\infty}|\hat{f}(n)|^{2}n^{2b}\log_{2}^{2}(n+1)<\infty$ for some
number $0\leq b\leq 1/2$ cannot diverge everywhere on a subset of Fourier
dimension $>1-2b$. Or again, the Fourier series of a function $f\in
H^{p}(\mathbb{T})$ converges $\mu$-a.e. for every Borel probability measure
$\mu$ such that $|\hat{\mu}(n)|^{2}\leq K|n|^{-\beta}$ for some $\beta>1-2p$.

##  6 Remark for Gaussian random series

In the particular case of Gaussian random variables, we obtain that we can get
rid of the factor $\log_{2}^{2}(1+n)$ thanks to the Sudakov–Fernique lemma. We
have the following:

###### Theorem 2

Let $(a_{n})$ be a sequence of real or complex numbers and $(X_{n})$ be a
sequence of real or complex Gaussian identically distributed random variables
with zero mean. If

$\displaystyle\sum_{n,m=1}^{\infty}|a_{n}||a_{m}||\mathbb{E}(X_{n}\overline{X_{m}})|<\infty,$
(4)

then the series $\sum_{n=1}^{\infty}X_{n}$ converges almost surely.

  * Proof

We shall first prove the following inequality:

$\displaystyle\mathbb{E}\left(\sup_{1\leq n\leq
N}|X_{1}+X_{2}+\ldots+X_{n}|\right)\leq
2\left(\sum_{n,m=1}^{N}|\mathbb{E}(X_{n}\overline{X_{m}})|\right)^{1/2}.$ (5)

Let $Z_{1},Z_{2},\ldots,Z_{N}$ be i.i.d Gaussian random variables with zero
mean and unit variance. Consider the sequence of random variables $(Y_{n})$,
$1\leq n\leq N$, given by

$Y_{n}=\left(\sum_{j=1}^{N}|\mathbb{E}(X_{n}\overline{X_{j}})|\right)^{1/2}Z_{n}.$

Clearly for all $1\leq n\leq m\leq N,$

$\mathbb{E}(|X_{n}+X_{n+1}+\ldots+X_{m}|^{2})\leq\mathbb{E}(|Y_{n}+Y_{n+1}+\ldots+Y_{m}|^{2}).$

Then by the classical Sudakov-Fernique inequality

$\mathbb{E}\left(\sup_{1\leq n\leq
N}|X_{1}+X_{2}+\ldots+X_{n}|\right)\leq\mathbb{E}\left(\sup_{1\leq n\leq
N}|Y_{1}+Y_{2}+\ldots+Y_{n}|\right).$

Now since the variables $(Y_{n})$ are independent, it is well-known that

$\mathbb{E}\left(\sup_{1\leq n\leq N}|Y_{1}+Y_{2}+\ldots+Y_{n}|\right)\leq
2\mathbb{E}\left(|Y_{1}+Y_{2}+\ldots+Y_{N}|\right).$

Since obviously $(\mathbb{E}|X|)^{2}\leq\mathbb{E}(|X|^{2})$, this yields,

$\mathbb{E}\left(\sup_{1\leq n\leq N}|X_{1}+X_{2}+\ldots+X_{n}|\right)\leq
2\left(\mathbb{E}\left(|Y_{1}+Y_{2}+\ldots+Y_{N}|^{2}\right)\right)^{1/2}.$

Hence

$\mathbb{E}\left(\sup_{1\leq n\leq N}|X_{1}+X_{2}+\ldots+X_{n}|\right)\leq
2\left(\sum_{k,j=1}^{N}|\mathbb{E}(X_{k}\overline{X_{j}})|\right)^{1/2}.$

Now fix a real number $r>0$ and $m\in\mathbb{N}$. Then

$\displaystyle\mathbb{P}\left(\sup_{1\leq
j<\infty}|X_{m}+X_{m+1}+\ldots+X_{m+j}|^{1/2}>r\right)$ $\displaystyle\leq$
$\displaystyle\frac{1}{r^{2}}\mathbb{E}\left(\sup_{1\leq
j<\infty}|X_{m}+X_{m+1}+\ldots+X_{m+j}|\right)$ $\displaystyle\leq$
$\displaystyle\frac{2}{r^{2}}\left(\sum_{k,j=m}^{\infty}|\mathbb{E}(X_{k}\overline{X_{j}})|\right)^{1/2}.$

Since $\sum_{k,j=1}^{\infty}|\mathbb{E}(X_{k}\overline{X_{j}})|<\infty$, then

$\lim_{m\to\infty}\sum_{k,j=m}^{\infty}|\mathbb{E}(X_{k}\overline{X_{j}})|=0.$

Hence

$\lim_{m\to\infty}\mathbb{P}\left(\sup_{1\leq
j<\infty}|X_{m}+X_{m+1}+\ldots+X_{m+j}|^{1/2}>r\right)=0.$

Since $r$ can be taken arbitrary small, this yields

$\lim_{m\to\infty}\mathbb{P}\left(\sup_{1\leq
j<\infty}|X_{m}+X_{m+1}+\ldots+X_{m+j}|^{1/2}>0\right)=0.$

This implies by an application of Fatou’s lemma (as in Kahane [5, p 30]) that
the series $\sum_{n=1}^{\infty}X_{n}$ converges almost surely.

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