11institutetext:  Information Systems Laboratory, FORTH-ICS, Heraklion, Greece
22institutetext: Computer Science Department, University of Crete, Heraklion,
Greece
22email: {fafalios, marketak, axaridou, tzitzik<EMAIL_ADDRESS>
#  A Workflow Model for Holistic Data Management and Semantic Interoperability
in
Quantitative Archival Research

Pavlos Fafalios 11 Yannis Marketakis 11 Anastasia Axaridou 11
Yannis Tzitzikas 1122 Martin Doerr 11

###### Abstract

Archival research is a complicated task that involves several diverse
activities for the extraction of evidence and knowledge from a set of archival
documents. The involved activities are usually unconnected, in terms of data
connection and flow, making difficult their recursive revision and execution,
as well as the inspection of provenance information at data element level.
This paper proposes a workflow model for holistic data management in archival
research; from transcribing and documenting a set of archival documents, to
curating the transcribed data, integrating it to a rich semantic network
(knowledge graph), and then exploring the integrated data quantitatively. The
workflow is provenance-aware, highly-recursive and focuses on semantic
interoperability, aiming at the production of sustainable data of high value
and long-term validity. We provide implementation details for each step of the
workflow and present its application in maritime history research. We also
discuss relevant quality aspects and lessons learned from its application in a
real context.

##  1 Introduction

Archival research is a type of research which involves investigating and
extracting evidence from archival records usually held in libraries, museums
or other organisations. In its most classic sense, archival research involves
the study of historical documents, thus it lies at the heart of original
historical research (Ventresca and Mohr,, 2017).

A large body of research in the field concerns the study of archival documents
that have a repetitive structure, such as registers, logbooks, payrolls,
censuses, etc., and which provide information about one or more types of
entities, such as persons, locations, objects, organisations, etc. Research in
this case usually starts by first collecting a set of archival documents
related to a domain of interest, which are then transcribed and curated for
enabling quantitative (but also qualitative) analysis of empirical facts,
their description and interpretation of possible causes, influences and
evolution trends (Petrakis et al.,, 2020).

Common data management problems in this context include: What data to
transcribe and how? How to curate the transcribed data for enabling valid
quantitative analysis and more effective exploration services? How to
integrate the data under a common schema/model for supporting the
investigation of information needs that require combining data from more than
one source? How to support the long-term preservation and reuse of the data?
How to maintain all data provenance information, which is important for the
verification and the long-term validity of research findings that use the
data?

Consider, for instance, the real use case of the SeaLiT
project111https://sealitproject.eu/ (ERC Starting Grant in the field of
maritime history), which studies the transition from sail to steam navigation
and its effects on seafaring populations in the Mediterranean and the Black
Sea (1850s-1920s) (Delis,, 2020). Historians in this project have collected
and studied a large number of archival documents of different types and
languages, such as crew lists, payrolls, and sailor registers, gathered from
multiple authorities in five countries. Complementary information about the
same entity of interest, such as a ship or a sailor, may exist in different
archival documents. For example, for the same ship, one source (accounts book)
may provide information about its owners, another source (naval ship register
list) may provide construction details and characteristics of the ship
(length, tonnage, horsepower, etc.), while other sources (crew lists) may
provide information about the ship’s voyages and crew. There might also be
another source (civil register) that provides additional information about the
crew members, such as their marital status and previous professions. Data
integration is very important in this context, for supporting historians in
finding answers to questions that require combining information from more than
one source, such as “finding the nationality of sailors of large ships that
arrived at a specific port”.

In addition, the name of the same entity (e.g. of a person) might be different
in different sources due to typos, different language, unrecognisable
characters, or use of abbreviation (e.g. ‘G. Schiaffino’, ‘Gaetano
Schiaffino’, ‘Gaetano Schiafino’). Moreover, the same term, such as a
profession or a ship type, may appear under different names in different
sources (e.g. ‘brigantine’, ‘brigantino’). Data curation, in particular entity
(instance) matching and term alignment, is crucial in this context for
enabling valid quantitative analysis (like grouping a list of retrieved
sailors by profession). However, at the same time, such curation must not
alter the original transcribed data since this is important for verification
and thus the long-term validity of the research findings.

To cope with these problems, in this paper we describe a workflow model for
holistic data management in archival research (depicted in Fig. 1). The
workflow relies on the strong collaboration between researchers (domain
experts) and data engineers (modeling experts), and focuses on semantic
interoperability, the ability of computer systems to exchange data with
unambiguous/shared meaning (Ouksel and Sheth,, 1999), because such an approach
supports the production of sustainable data of high value that can be extended
and re-used beyond a particular research activity or project.

The workflow was designed based on real users’ needs and is provenance-aware,
in the sense that it retains the full provenance chain of each piece of data.
It achieves this by decoupling data entry from data curation and integration.
The researcher can go back to the transcript or the original source and
inspect the initial form of a piece of information. It is also highly-
recursive, supporting the revision of the transcription, curation and
integration steps, e.g. due to new knowledge acquired in the course of
research. In comparison to related work, we treat the relevant activities in
an holistic manner, paying particular attention on maintaining the provenance
information at micro (data element) level, which is important for reproducible
research in the age of Open Science (Vicente-Saez and Martinez-Fuentes,,
2018).

We showcase an implementation of the workflow model in a real use case in the
field of maritime history and report empirical results from its application
for satisfying real information needs of a large group of historians. We also
discuss relevant data quality aspects and lessons learned.

The rest of this paper is organised as follows: Section 2 provides the
required background and describes related work. Section 3 provides an overview
and the main characteristics of the proposed workflow model. Section 4 details
how each step of the workflow model can be realised. Section 5 provides
information about the automation of the workflow. Section 6 describes a real
use case. Section 7 discusses quality aspects and relevant lessons learned.
Finally, Section 8 concludes the paper and outlines future work.

##  2 Background and Related Work

We first explain the basic notions about semantic technologies (Section 2.1)
and review how such technologies are used in humanities research, a large part
of which concerns archival research (Section 2.2). We then focus on the
different data management activities towards semantic interoperability in
archival research and present relevant works (Section 2.3). Finally, we
position our work (Section 2.4).

###  2.1 Basic Notions

Semantic technologies aim at helping machines understanding data. RDF
(Resource Description Framework)222https://www.w3.org/TR/rdf11-concepts/ and
OWL (Web Ontology Language)333https://www.w3.org/TR/owl2-overview/ are key
semantic technologies that enable encoding the semantics of data, thus
allowing to formally represent the meaning involved in information (Antoniou
and Van Harmelen,, 2004). This representation has the form of a semantic
network (or knowledge graph) which stores interlinked descriptions of
“entities” (objects, persons, events, concepts, etc.) in a graph structure in
which vertices represent entities and edges represent semantic relations
between the entities. Typical standardized semantic networks are expressed as
RDF triples (statements of the form subject-predicate-object) stored in a
semantic repository (RDF triplestore) (Ali et al.,, 2021). Semantic
technologies help achieving semantic interoperability, the ability of computer
systems to exchange data with unambiguous/shared meaning, which is a
requirement to enable machine computable logic, inferencing, knowledge
discovery, and data federation between information systems (Ouksel and Sheth,,
1999).

###  2.2 Semantic Technologies for Humanities Research

There is an increasing adoption of semantic technologies in the humanities
field, with a main focus on how to produce and make publicly available
interoperable Linked Data (Heath and Bizer,, 2011) that can be easily queried
and integrated with other datasets (Hyvönen,, 2020; Hyvönen et al.,, 2014;
Hawkins,, 2021; Beretta,, 2021; Fafalios et al., 2021a, ).

Oldman et al., (2015) provide a critical discussion on how semantic
technologies and the idea of Linked Data are used in humanities research, and
describe strategies for the wider adoption of these technologies for
supporting high-quality digital humanities projects and the production of data
that better represents human knowledge and better reflects the needs of
humanities researchers. Hawkins, (2021) examines how Linked Data about
archives is beneficial for those engaged in digital humanities research and
scholarship, considering some of the barriers that currently prevent digital
humanists from being able to utilise digitised and born-digital archives.

We believe that the workflow model that we propose, in particular its
provenance-awareness at data element level, is a first step towards tackling
some of the major issues described in the aforementioned works, such as the
ability “to trace the provenance of knowledge back to the source micro‐level
(with its original context and perspective intact)” (Oldman et al.,, 2015,
p.10), or “preventing the decontextualisation and loss of nuance of archives”
(Hawkins,, 2021, p. 11).

With respect to historical research, for which archival research is a core
part, Meroño-Peñuela et al., (2015) survey the joint work of historians and
computer scientists in the use of semantic technologies. The article provides
an extensive analysis on works and systems for knowledge modelling, text
processing and mining, search and retrieval, and data integration. It also
discusses aspects of semantic technologies that could be furtherly exploited
in historical research. Such an aspect is the “non-destructive data
transformations” (Meroño-Peñuela et al.,, 2015, p. 22). Decoupling data entry
from data curation and transformation, and maintaining a recursive workflow
between these processes, are core characteristics of the proposed workflow
model that help towards this direction.

###  2.3 Data Management for Semantic Interoperability in Archival Research

Common data management activities for enabling semantic interoperability in
archival research include:

  * •

digitization / transcription of archival documents (scanning of documents,
text recognition, manual transcription)

  * •

documentation / metadata recording (what is the origin of a document, what is
the document about, who makes the transcription, etc.)

  * •

data curation / preparing the data for statistical analysis (correction or
normalisation of data values, instance matching, term alignment, etc.)

  * •

data integration under a common representation language (ontology-based
modeling, creation of mappings, data transformation)

  * •

data publication (e.g. as Linked Data)

  * •

data analysis and exploration (qualitative and/or quantitative analysis, query
building, data visualisation, etc.)

There is a plethora of software tools and systems for each of these
activities. Below we present relevant works that have a focus on humanities
research.

Digitization/Transcription. One can either use text recognition software for
automatically extracting text from historical documents, or manually perform
the transcription process, each approach having its pros and cons. For
example, the automated approach usually needs large amounts of training data
and its effectiveness (quality of results) highly depends on the kind/quality
of text to be extracted and the amount of training data. On the other hand,
manual transcription provides high quality results but it requires a lot of
effort. A mixed method is to combine automated extraction with manual
correction and data entry. Regarding software tools, Transkribus (Kahle et
al.,, 2017) is a popular platform for the digitisation of historical
documents, offering AI-powered text recognition. FastCat (Fafalios et al.,
2021b, ) is a web application for manual and collaborative transcription based
on templates. It organises the data (and metadata) in tabular forms (tables),
similar to spreadsheets, offering a fast and user-friendly way to data entry.

Documentation / metadata recording. There are two main approaches for
documentation towards semantic interoperability: a) decoupling the
documentation process from the ontology-based integration and the production
of the semantic network, b) creating the semantic network from the very
beginning, i.e. during the documentation process. Synthesis (Fafalios et al.,
2021a, ) is a web-based system that applies the first approach for the
collaborative and scientific documentation of cultural entities (objects,
events, persons, organisations, etc.), offering embedded processes for
transforming the data to an ontology-based RDF dataset. ResearchSpace (Oldman
and Tanase,, 2018) and WissKi (Scholz and Goerz,, 2012) are platforms that
apply the second approach, supporting the direct ontological representation of
(meta)data.

Spreadsheet software, such as Microsoft Excel, and relational database
management systems (RDBMS), like Microsoft Access, are still popular (and
probably the dominant) tools for (meta)data entry and analysis, and are
extensively used for manual documentation and metadata recording. There are
also RDBMS-based systems, such as HEURIST444http://heuristnetwork.org/ and
nodegoat555https://nodegoat.net/, that are tailored to humanities researchers
and which combine a set of functionalities for building and managing research
datasets, without however focusing on semantic interoperability.

Data curation. This is an optional step which is usually undertaken when a
quantitative (statistical) analysis of the transcribed data is needed. In such
a case, curation is very important because data quality can affect the
reliability of the analysis results. OpenRefine666https://openrefine.org/ is a
popular desktop application for data cleaning. It operates on rows of data
which have cells under columns (similar to relational tables).
Silk777http://silkframework.org/ (Volz et al.,, 2009) is an open source
framework for finding links between related data items, e.g. for instance
matching. It provides a declarative language for specifying linkage rules and
support of RDF link generation, through owl:sameAs or other types of links.
For fully-automated instance matching (entity resolution), there is a plethora
of learning-based methods that require manually or automatically generated
training data (Christophides et al.,, 2020). Finally, the FastCat system
(Fafalios et al., 2021b, ) offers a web-based environment, called FastCat
Team, which supports both automated (rule-based) and manual instance matching
and vocabulary curation processes. The applied curation does not alter the
original (transcribed) data and maintains links from the curated to the
original data.

Data integration. The objective here is to semantically represent all data and
metadata using a domain (formal) ontology (as the common representation
language), in order to enable semantic interoperability and make the data
exploitable beyond a particular research problem or project. This activity
includes the data modeling and data transformation processes. Data modeling
consists of defining or selecting the domain ontology and creating the schema
mappings, while data transformation transforms the data based on the schema
mappings and creates the semantic network of integrated data.

Regarding software systems, Protégé is a popular ontology editor which
provides a graphic user interface to define ontologies. It can be used for
creating a new ontology for a given domain in OWL, or for building an
extension of an existing ontology. For the creation and execution of schema
mappings, R2RML888https://www.w3.org/TR/r2rml/ is a W3C standard for mapping
relational databases into RDF, while Dimou et al., (2014) describe an
extension called RML for mapping heterogeneous sources into RDF. Finally, the
X3ML toolkit (Marketakis et al.,, 2017) provides a declarative (XML-based)
mapping definition language as well as a set of tools for the creation and
maintenance of the schema mappings, and the actual transformation of the data
to RDF.

Data publication The integrated data can be now imported in a semantic
repository (RDF triplestore), either publicly available or private, which
offers an Application Programming Interface (API) for accessing the data and
running structured queries using the
SPARQL999https://www.w3.org/TR/sparql11-overview/ protocol and language. Then,
user-friendly applications can be built on top of this API for supporting end
users in exploring and analysing the integrated data. The data can be also
published as Linked Data, following the Linked Open Data (LOD) principles
(Heath and Bizer,, 2011). The Sampo
model101010https://seco.cs.aalto.fi/applications/sampo/ (Hyvönen et al.,,
2014) provide a framework for collaborative publishing and using of LOD, which
has been tested in several domains by building the so-called ‘Sampo portals’
(Hyvönen and others,, 2020).

Data exploration and analysis. There are two main general methods that can be
used for exploring the integrated data: (a) free text search: the user
provides a set of keywords or a natural language question, as in ad-hoc
information retrieval, (b) interactive interface: the user is supported by the
system to express an information need, through a user-friendly interactive
interface. In both cases the result is (usually) a ranked list of entities
from which the user can start exploring relevant information, e.g. through
browsing, faceted search, or different visualisations such as charts, maps,
timelines, etc.

There is a plethora of different methods for implementing keyword search over
RDF data, e.g. using a document-centric information retrieval system
(Kadilierakis et al.,, 2020), or by translating a keyword query to a
structured (SPARQL) query (Izquierdo et al.,, 2021). For the presentation of
the keyword search results, Nikas et al., (2020) suggest a multi-perspective
approach that offers multiple presentation methods (perspectives), allowing
the user to easily switch between these perspectives and thus exploit the
added value of each one. Regarding interactive interfaces, A-Qub (Kritsotakis
et al.,, 2018) and ResearchSpace (Oldman and Tanase,, 2018) offer user-
friendly environments which support end users in gradually building complex
questions (corresponding to SPARQL queries) that associate different types of
entities and information.

###  2.4 Positioning

To the best of our knowledge, there is no related work that approaches the
data management part of archival research in an holistic manner, in the sense
that the proposed workflow model enables the representation and efficient
management of information, applies semantic data integration facilities in
order to provide a rich knowledge graph of archival data, and at the same time
it preserves the full provenance chain allowing researchers traverse from the
final semantically integrated collection back to the original and transcribed
manuscripts and vice versa.

##  3 Workflow Model: Overview and Main Characteristics

We first provide an overview of the workflow model (Section 3.1) and then
highlight its distinctive characteristics (Section 3.2).

###  3.1 Roles, Input/Output and Processes

Fig. 1 depicts the proposed workflow model for supporting holistic data
management in archival research.

Figure 1: Workflow model for holistic data management and semantic
interoperability in archival research.

Roles. There are two main roles engaged in the workflow: a) the researcher
(domain expert / end user), who collects and studies the archival material,
provides domain knowledge, and defines requirements, and b) the data engineer
(modeling expert), who designs and implements the different workflow
processes.

Input/Output. The input of the workflow is a set of archival documents
gathered from different authorities by one or more researchers, together with
information needs provided by the researchers that are related to their
research aims and for which the gathered archival material can provide
important information (evidence). The gathering of information needs is very
useful in this stage because it allows data engineers to better design and
implement the next workflow processes. The output of the workflow is a rich
semantic network (a knowledge graph) of integrated information, which is used
by the researchers for data analysis and exploration, as well as two distinct,
intermediate databases: a database of records (original transcripts), and a
database of curated data (curated entity instances and vocabulary terms).

Process 1: Creation of Source Schemas. Following the description logic based
framework for information integration as introduced by Calvanese et al.,
(1998), we first need to create the source schemas, one for each different
type of source, which provide the required data entry forms in a software
system for the transcription and documentation of the original archival
documents. This first step enables data curation and consolidation relative to
source model semantics, as well as modeling and integration under a common
ontology which can be modified in the course of research, without this
affecting/delaying the transcription process. The close collaboration between
the researchers and the data engineers is very important in this process for
properly designing the schemas and avoiding mistakes during data entry that
can cause difficulties/limitations in the next steps. An example of such a
mistake is the use of a single data entry field for the recording of a
measurement unit and value. This is very likely to cause issues to the end
user when wanting to perform comparisons during data exploration.

The creation of a new source schema, or a modification/extension of an
existing one, will be required if new archival documents of a different type
of source are gathered by the researchers and need to be transcribed. This can
happen at any stage of the overall pipeline and does not affect the other
processes that can run in parallel for the existing gathered material.

Process 2: Transcription. After having created one or more source schemas for
the gathered archival documents, the transcription of the documents can begin
by the researchers using a software system that offers the required data entry
forms. Apart from the transcription of the important document contents, this
step includes the recording of metadata information for both the documents
(archive/library, dating, etc.) and the transcription process (who makes the
transcription, etc.). The result of the transcription process is a database of
transcripts. This is a task solely performed by the group of researchers, but
which can make use of software tools for facilitating/automating
transcription, such as text recognition software.

Process 3: Curation. The next step of the workflow is the curation of the
transcribed data. At this stage researchers need to harmonise the different
data elements that appear in the transcripts and resolve identity ambiguities,
so that different elements that co-refer to the same real-world entity/concept
receive the same identifier, and false co-references are disassociated.

The data elements can be divided into two main categories: (a) universals;
concept instances that belong to a specific vocabulary or thesaurus of terms,
such as professions, object types, etc., and (b) particulars; entity instances
that belong to specific categories and are accompanied by
characteristics/properties, such as persons (first name, last name, birth
date, etc.), locations (name, type, etc.), organisations (name, location,
etc.). Curation can also include the provision of corrected/preferred values
(e.g. correcting the first name of a person instance) or the entity enrichment
(e.g. adding coordinates to a location instance), tasks which are usually
important for better data exploitation and visualisation by the external
services that operate over the curated and integrated data.

The curation process is a task performed by the group of researchers and may
include both manual and automated steps. For example, instance matching of
entities, or alignment of vocabulary terms, can comprise both an automated
step (based on rules) and a manual step (for validation of ambiguous cases).
The result is a distinct database of curated data, with links to the original
data elements, which means that the curation step does not alter the data as
transcribed from the original sources.

Process 4: Ontology-based Data Integration. The next step is the ontology-
based integration of the transcribed and curated data, which includes the
modeling and transformation sub-processes.

For modeling, the good practice suggests to either use an established domain
model (if such a model is available for the application domain), or create a
new model (a specialised extension) that is compatible to an established upper
ontology. This process usually requires extensive discussions between the
domain experts, who know the data, and the data engineers, who build the
domain ontology and create the mappings.

An important part of the modeling process is the creation of the schema
mappings that describe how the input data (transcripts and curated data) are
mapped to classes and properties of the domain ontology. In general, the
creation of the schema mappings can be a time-consuming process when the
source schemas are many and large/complex. Nevertheless, it needs to be done
only once for each different type of source, while revisions may be required
if there are changes in the schemas or the target ontology. The use of a
declarative language for defining the mappings, such as X3ML (Marketakis et
al.,, 2017), is recommended because local changes in the sources require local
changes in the mapping specifications that are easy to locate and perform.

The transformation process takes as input i) the databases (outputs of
transcription and curation processes), ii) the domain ontology, and iii) the
schema mappings, and produces a rich semantic network of integrated data. This
step can be fully automated and can repeated for any new data sources that are
transcribed and curated, as long as there is no change in the transcription
schemas.

Process 5: Research, analysis, exploration. The resulting semantic network of
integrated data is exploited by the researchers through one or more services
that operate over the semantic network and which offer user-friendly
interfaces for data browsing, analysis, and exploration. Here it is important
for the end users to be able to go back to the transcripts, or even the scans
of the original sources, for inspecting the initial form of a piece of
information (before its curation and transformation), or for gathering further
contextual information. In addition, in the course of research, a user may
identify that corrections are needed in the transcribed or curated data, thus
researchers need to be able to revisit the transcription and curation steps,
make corrections, and then re-transform (automatically) the data for updating
the semantic network. Likewise, new archival documents might be collected at
any time, which means that one or more new source schemas and corresponding
mappings might need to be created for enabling their transcription, curation
and transformation.

###  3.2 Workflow Distinctive Characteristics

Below we highlight and motivate the distinctive design and methodological
characteristics of the proposed workflow model:

  * •

Strong collaboration between researchers (domain experts) and data engineers
(modeling experts). Such a collaboration is required for a) better designing
the source schemas (and the corresponding data entry forms), b) better
defining/designing the target (domain) ontology and creating the schema
mappings, and c) better creating/configuring the user interfaces of the data
exploration service(s).

  * •

Decoupling data entry from data curation and maintaining links from the
curated to the original data. This is very important not only for maintaining
the data provenance, verifying information, and thus validating the research
findings that make use of the data, but also because data curation and
consolidation may be ambiguous and require further research and repeated
revision at any time in the future (by the same or other researchers).

  * •

Separating source schema creation from ontology modeling. We aim at removing
the bias of the initial research hypothesis from the target (integration)
model, one of the most severe philosophical problems of unbiased research and
at the core of the discussion about scientific realism (Turner,, 2007; Chapman
and Wylie,, 2018). The target model (ontology) can be developed in parallel
with the data entry process and can be re-adapted at any time to new insight
from the sources, without invalidating the entered data and without this
affecting (or delaying) the transcription and curation processes.

  * •

Separating the databases (of transcripts and curated data) from the semantic
network. Decoupling data entry and curation from the creation of the semantic
network enables maintaining the semantics of the source model by keeping the
transcripts as close to the original (archival) document as possible (trying
to maintain their original structure), offering at the same time a familiar
way to data entry that can highly speed up this time consuming process. In
addition, this allows the straightforward production of different versions of
the semantic network, considering different ontologies, or different versions
of the same ontology (this only requires creating the schema mappings based on
the desired target model).

##  4 How to Realise the Workflow

We now provide implementation details for realising the workflow.

###  4.1 Faithful, Fast and Collaborative Data Transcription

Common requirements that a data transcription system should satisfy, include:

  * •

Supporting the faithful and structured transcription of information from the
archival documents (as exact to the original information as possible), as well
as the recording of metadata information.

  * •

Supporting fast data entry through an intuitive user interface that
researchers are familiar with or can quickly get familiar with.

  * •

Supporting the collaborative transcription by more than one researcher, making
use of the same structures (source schemas) for data entry.

These characteristics can highly affect the usability of the data entry system
and thus its acceptance by the end users (researchers).

For enabling the next data curation process, we first need to identify what
are the main entity categories (like persons, locations, objects, etc.) and
the main vocabularies or hierarchies of terms that appear in the transcribed
data and need curation. To this end, we need to define the fields in the data
entry forms that provide entity or term related information. For example, the
data entry fields first name and last name provide information for a person
instance, while the field profession provides a vocabulary term. The values of
these fields must be copied (ideally, automatically) to a new environment that
allows for their curation without altering the original data as it appears in
the transcripts. We then only need to provide a link from the curated to the
original data and/or position information (e.g. record name, table name, row
number), in order to retain the provenance information.

###  4.2 Provenance-aware Data Curation

Data curation activities that need to be supported by a dedicated software
system include:

  * •

Correcting the name of an entity or the value of one of its properties (by
setting a preferred label).

  * •

Instance matching: matching two or more entity instances that refer to the
same real-world entity, which means that they must receive the same identity.

  * •

Instance unmatching: unmatching a specific entity instance from a set of
automatically matched instances, which means that the instance will receive a
different identity.

  * •

Enrichment: complementing an entity instance with additional information, like
adding coordinates to a location.

  * •

Providing a preferred term for a vocabulary term (e.g. a term from a fixed
thesaurus, or a term in English for a term in another language).

  * •

Providing a broader term for a vocabulary term (thereby creating an hierarchy
of terms).

Instance matching in this context can be multi-stage. A first automated step
can assign the same identity to a set of entity instances having some common
characteristics, e.g. common first name, last name, and birth date, in the
case of person instances (rule-based approach), or make use of machine
learning techniques (supervised or semi-supervised approach) (Christophides et
al.,, 2020). Then, a second manual step (performed by the researchers) can
match additional entity instances that the automated step did not manage to
match, or unmatch an entity instance that was incorrectly matched to other
instances by the automated step.

The instance matching/unmatching activities and the provision of preferred
terms for vocabulary terms are of key importance for valid quantitative
(statistical) analysis over the integrated data. Consider, for example, that a
researcher who studies archival documents related to maritime history (like
crew lists) wants to find the birth place of sailors that arrived at a
specific port, or group them by their profession. Providing the same identity
to all sailor instances that represent the same real-world person, as well as
providing the same ‘preferred’ term for all different professions that
correspond to the same profession, ensures that the generated aggregated
information is correct.

###  4.3 Ontology-based Integration

The ontology-based integration of the transcribed and curated data consists of
the below tasks:

  1. 1.

Data modeling using a domain ontology.

  2. 2.

Creation of schema mappings and definition of how to generate the entity
identifiers (URIs).

  3. 3.

Running the transformations for producing the semantic network of integrated
data.

Data modeling. CIDOC-CRM111111http://www.cidoc-crm.org/ (Doerr,, 2003) is a
high-level, ISO standard ontology (ISO
21127:2014)121212https://www.iso.org/standard/57832.html of human activities,
things and events happening in space and time, thus it can be used for
modeling the transcribed data and supporting semantic interoperability and
long-term data preservation. Depending on the application domain, an extension
of CIDOC-CRM might be required for specialising particular notions of
interest. For instance, in our use case we created the SeaLiT Ontology, an
extension of CIDOC-CRM for the modeling and integration of data related to
maritime history (more in Section 6). For semantic data management using
CIDOC-CRM, Tzitzikas et al., (2022) analyse the relevant processes and tasks,
and review the literature on applying machine learning techniques for reducing
the costs related to compliance and interoperability based on CIDOC-CRM.

Mapping & Generation of Identifiers. This step defines how the transcribed and
curated datasets will be transformed so that they will eventually construct
the semantic network. The challenge is to preserve the full provenance chain,
from the curated data to the original data of the transcript of the source, so
that researchers can easily validate, further improve, or seek for further
information. The first part of this step is the definition of the schema
mappings, that identify which parts from the input schema (e.g. a particular
table column) will be mapped to concrete classes and properties of the domain
ontology, ensuring that the semantics of the original data are well-defined,
non-ambiguous, and no data is lost. The second part defines how resource URIs
and labels will be generated. At this point URIs will be used as the ‘glue’
connecting relevant pieces of information.

Fig. 2 shows an indicative example on how URIs are used for establishing such
connections. In this example there are two different transcription records,
each one of them describing various persons. In one of them there is a person
called ‘Agostino B??ndi’ (i.e. the question marks reveal that the characters
could not be recognised from the original source), and in another one there is
a person called ‘A Brondi’. For these persons two different URIs are created,
since their names do not match and also they are found in different records.
However in the curated dataset, historians agreed that these references point
to the same person. Therefore a new person instance is created, with a new
URI, linked to the previous ones. This new instance is called ‘master’, while
the linked instances are considered ‘local’.

Figure 2: Identity (URI) management and provenance chain.

Each URI consists of three parts: (a) the URI prefix which is common for all
the resources, (b) the type or hierarchy of the resource, (c) the actual or
hashed content of the resource. An indicative URI is:
https://rs.sealitproject.eu/kb/location/ sardinia. We should also mention,
that there are cases where the aforementioned strategy is not applied. An
indicative case is the construction of intermediate nodes in the semantic
network, for which a URI is not required (e.g. the ‘E67 Birth’ event). In such
cases a random UUID is assigned for them.

Transformation. This step takes as input (a) the transcribed and curated
datasets and (b) the definitions of the schema mappings and URI generators,
and produces the ontological instances (RDF triples) with respect to the
domain ontology, that are the core contents of the semantic network. This step
does not require any human intervention and can be fully automated. One
apparent advantage of this automation is that the semantic network can be
fully or partially refreshed as soon as new data have been transcribed and/or
researchers have curated more data.

A good practice for managing the semantic data in terms of updating and
versioning flexibility is the use of named graphs (Carroll et al.,, 2005), one
for each source record. When there is a new version of a record, or of its
mapping definition file, the record output produced with a new workflow cycle
can be easily integrated in the semantic repository by replacing the RDF data
in the corresponding named graph. Also, the hierarchies of terms and locations
can be effectively managed and updated in distinct named graphs, as well as
the result of the materialisation process for semantically inferred statements
(the production of new RDF triples as shortcuts that represent long paths, for
improving query performance).

###  4.4 Semantic Network Exploitation

The integrated data of the semantic network can be now exploited as a primary
source for archival research. This includes finding answers to complex
information needs and analytical queries that require combining information
from different sources, as well as visualising the results in different forms,
such as tables, charts, timelines, or maps, for direct use in research.

The actual information needs depend on the application domain and the type of
exploration or analysis needed by the end users. The challenge here is to
provide researchers with user-friendly and intuitive-to-use interfaces that
they can trust for expressing their information needs and findings relevant
information. Thus, the key success factors of such data exploration services
are usability and trustworthiness. The latter can be achieved by enabling
users to directly inspect the provenance of the displayed information, by
allowing them to directly visit the transcript containing the information, or
even a scan of the original archival document.

Some general categories of information needs include: (i) finding information
about a particular entity, such as the birth date and place of a person; (ii)
retrieving a list of entities based on one or more properties of these
entities (e.g. all persons having a specific residence location); (iii)
grouping a list of retrieved entities based on some property or characteristic
(e.g. grouping all retrieved persons by their profession); (iv) finding
comparative information related to some entities (e.g. number of persons
employed by the organisation in different time periods).

Finally, a strategy on how to handle missing values in the data, which is very
common for certain types of archival documents, is very important in order to
get valid aggregated information and make safe conclusions. For example, the
residence location for some persons might be empty in the original document.
When grouping a set of persons by their residence location, there must be an
‘unknown’ value for this missing information.

##  5 Workflow Automation

The systems used in the transcription and curation processes need to
intercommunicate for automating the copy of the data elements (entities,
terms) that need curation from the transcription system to the curation
system. Then, an important part of the workflow can be fully automated as long
as the modeling process has been completed and the mappings for all different
source schemas have been created (tasks that need to be done once for each
different type of archival documents). In this case, new transcribed and
curated data can be automatically transformed and imported in the semantic
repository of integrated data, and thus directly be explored by the end users
through the data exploration application.

Specifically, the workflow scenario is the following: a group of researchers
have collected a first set of archival documents and the data entry forms have
been created in a dedicated system for each different type of source. The
researchers start the transcription process. When transcription has been
completed for the collected set of archival documents, the data elements that
need curation are automatically copied to the curation environment and
researchers start curating them. At the same time, data engineers, with the
support (domain knowledge) of the researchers and by studying the available
material evidence and the experts’ requirements, define the target (domain)
ontology and create the schema mappings for each different type of source.
When both the transcription and curation processes have been completed for all
(or a large set) of the archival documents, and the corresponding schema
mappings have been created, researchers can ‘publish’ the data, which means
that the transformation process is executed and the semantic network is
created and ingested in a semantic repository. Researchers can then start
exploring the integrated data through the user-friendly interface of an
application that operates over the semantic repository.

At any time, researchers can transcribe and curate new archival documents, or
make corrections in the existing (curated) data due to new knowledge acquired
in the course of research, and then re-execute the transformation process and
update the semantic repository automatically. The changes in the semantic
repository are directly (and automatically) reflected in the data exploration
application.

The entire set of archival documents to be considered by the researchers does
not need to be known from the beginning, meaning that new documents might be
collected for transcription at any time. In this case, creation of new source
schemas (data entry forms) is needed if such new documents belong to a new
type of source which is different from the existing ones. Accordingly, changes
in an existing data entry form might be needed (e.g. addition of a new column)
in order to enable the transcription of a new important type of information
that was not originally planned or known for an existing type of source. In
both cases, revision/extension of the domain ontology might be needed, as well
as creating new schema mappings or applying changes in the existing ones.

Note here that, even if there are changes in the transcription schemas and the
integration model, which actually occur during the course of a project, such
changes are independent of the other transcription and curation processes
performed (in parallel) by the researchers (thus, they do not affect or delay
them). Moreover, the full automation of the data transformation step reduces
the overhead for the researchers to the absolute minimum.

The two steps of the workflow that are the most time consuming are the
transcription and curation processes. As already stated, several sub-tasks in
these two processes can be automated or semi-automated, e.g. using state-of-
the-art text recognition software (Kahle et al.,, 2017), or applying automated
instance matching / entity resolution (Christophides et al.,, 2020). Here the
challenge is to find the best trade-off between fully automating the tasks and
having results of high accuracy for enabling valid data analysis. We suggest
semi-automated solutions that consider human-in-the-loop for ensuring high
quality (Wu et al.,, 2022; Gurajada et al.,, 2019).

##  6 Use Case in Maritime History Research

The workflow has been fully implemented in a real use case for supporting a
large number of historians in managing a diverse set of archival sources
related to maritime history. The context is the project
SeaLiT131313https://sealitproject.eu/, in which maritime historians study the
transition from sail to steam navigation and its effects on seafaring
populations in the Mediterranean and the Black Sea (1850s-1920s).

Below we provide details on how each process of the workflow was implemented
and illustrate an example on how a real information need provided by the
historians is satisfied by exploiting the integrated data.

Archival material and information needs. The archival material studied in
SeaLiT covers a variety of sources in five languages (Spanish, Italian,
French, Russian, Greek), including crew and displacement lists, registers of
different types (sailors, naval ships, students, etc.), logbooks, payrolls,
account books, employments records, and censuses. Details about the full
archival corpus and its origin is a available in the project’s web
site.141414https://sealitproject.eu/archival-corpus

Our first task was to gather a set of information needs from the historians of
SeaLiT, related to their research aims and for which the studied archival
material can provide important information. This is fundamental for better
designing the source schemas (data entry forms), the integration model, as
well as the data exploration services. We collected around 100 information
needs. Indicative examples are:151515The full list of gathered information
needs is available at
https://users.ics.forth.gr/~fafalios/SeaLiT_Competency_Questions_InfoNeeds.pdf

  * •

What are the places of construction of ships during a specific period?

  * •

What are the most popular European destinations (under a chronological
perspective) of the ships from the Black Sea?

  * •

How many people that arrived at a specific place (e.g. Barcelona) have place
of birth more than X miles away?

  * •

How many ship owners per ship during a specific period?

Creation of source schemas and transcription. The FastCat system (Fafalios et
al., 2021b, ), which is available as open source
software161616https://github.com/isl/FastCat, was used for the creation of the
source schemas and the transcription of the archival documents by around 30
users in 5 countries (historians of SeaLiT). In FastCat, users can transcribe
documents and provide metadata information by creating ‘records’ belonging to
specific ‘templates’. A ‘record’ organises the data and metadata of an
archival document in a set of tables, while a ‘template’ represents the
structure of a distinct data source, i.e. it defines the data entry tables,
their columns as well as the type of each column (for denoting columns that
provide vocabulary terms or entity-related information, whose values will be
curated after transcription). For the case of SeaLiT, twenty templates were
created, one for each different type of archival source. Table 1 provides the
templates as well as an overview of the information that can be recorded in
each template.

Table 1: Considered archival sources and overview of recorded information. Archival source |  Overview of recorded information
---|---
Crew and displacement list (Roll) |  Information about ships, crew members, ports.
Crew List (Ruoli di Equipaggio) |  Information about ships, voyages, crew members, ports.
General Spanish Crew List |  Information about ships, ship owners, crew members, voyages, ports.
Sailors Register (Libro de registro de marineros) |  Information about sailors (including profession and military service organisation locations)
Register of Maritime Personnel |  Information about persons (including residence location, marital status, previous profession, military service organisation locations).
Register of Maritime Workers |  Information about maritime workers, ships, captains, ports.
Seagoing Personnel |  Information about persons (including marital status, profession, end of service reasons), ships, destinations.
Naval Ship Register List |  Information about ships (including tonnage, length, construction location, registration location) and ship owners.
List of Ships |  Information about ships (including previous names, registry port and year, construction place and year, tonnage, engine characteristics, owners).
Civil Register |  Information about persons (including profession, origin location, marital status, death location and reason).
Maritime Register, La Ciotat |  Information about persons, embarkation and disembarkation locations, ships, captains, patrons.
Students Register |  Information about students and courses.
Census La Ciotat |  Information about occupants (including nationality, marital status, religion, profession, working organisation, household role).
Census of the Russian Empire |  Information about occupants (including marital status, estate, religion, native language, household role, occupation).
Payroll (of Greek Ships) |  Information about ships, captains, voyages, persons, employments (including wages).
Payroll (of Russian Steam Navigation and Trading Company) |  Information about ships, persons, recruitments (including salary per month).
Employment records (Shipyards of Messageries Maritimes, La Ciotat) |  Information about workers (including marital status, profession, status of service in company).
Logbook |  Information about ships, captains, ports, route movements, voyage events.
Accounts Book |  Information about ships, voyages, captains, ports, transactions.
Notarial deeds |  Information about deeds, notaries, witnesses, contracting parties, ships.

The total number of records transcribed by the historians of SeaLiT is
currently more than 620. Fig. 3 shows a part of a real record belonging to the
template Crew List (Ruoli di Equipaggio)171717The full record is accessible
at: https://tinyurl.com/2u35frya (there are totally 98 records belonging to
this template). This template consists of six tables, enabling historians to
provide/transcribe information about: i) the record itself (creation date,
last modification date, transcriber); ii) the source (archive/library,
location, register number, issuing authority, etc.); iii) the ship (name,
type, tonnage, construction location, etc.); iv) the crew list (embarkation
port and date, discharge port and date, surname, name, residence location,
profession, payment information, etc.); v) the documented navigation (date,
duration, first planned destination, total crew number); vi) the route
(departure port and date, arrival port and date). In the record of Fig. 3, for
instance, the transcriber has provided data for twenty six sailors and
thirteen route ports that concern the navigation of the the ship Pallade (type
Brigantino) from 11-01-1861 to 26-02-1862.

The creation and configuration of the templates in FastCat was not an ‘one
shot’ process. New templates were created periodically based on new archival
material gathered from the historians, or existing templates were changed
several times even after the creation of records (e.g. by including additional
columns in a table), for incorporating new (and important) type of information
provided by particular archival documents.

Figure 3: An example of a real FastCat record belonging to the template ‘Crew
List (Ruoli di Equipaggio)’.

Curation. The curation of the transcribed data (vocabulary terms and entity
instances) is performed through a dedicated environment within FastCat, called
FastCat Team. Specifically, when a historian has completed the transcription
of one or more documents (records), the record(s) can be ‘published’, which
means that all data concerning vocabulary terms and entity instances are
copied to FastCat Team for enabling their curation.

In the case of SeaLiT, the current number of vocabularies is fifty two
(examples include: ship type, engine type, profession, marital status), while
the types (and current number) of entities that can be curated are ships
(about 2,400), persons (about 99.200), locations (about 9,800), legal entities
(about 1,100). For each term in a vocabulary, the user can provide a preferred
term (in English) and a broader term, or inspect the records in which the term
appears. For the curation of the entity instances, the user can correct
values, select two or more instances for matching them (indicating that they
represent the same real-world entity), unmatch a particular instance from a
set of automatically-matched instances, or inspect the records in which the
entity instance appears. In the case of locations, the user is able to add an
identifier (TGN/Geonames ID), as well as coordinates or a secondary location
name (e.g. a historical name).

Fig. 4 shows the user interface of FastCat Team, in particular the page that
allows the curation of ship instances. For more information about FastCat (and
FastCat Team), the reader can refer to Fafalios et al., 2021b .

Figure 4: Curation of ship instances in FastCat Team.

Ontology-based integration and transformation. For data integration we created
a data model compatible with CIDOC-CRM, called ‘SeaLiT
Ontology’181818https://zenodo.org/record/6797750. The current version of the
ontology (v1.1) contains forty six classes and seventy nine properties,
allowing the description of information about ships, voyages, employments,
payments, seafaring people, teaching courses, and other relevant activities.
For creating the schema mappings and transforming the data to RDF we make use
of the X3ML framework (Marketakis et al.,, 2017). In particular, one mapping
definition file has been created for each template in FastCat, as well as one
for each of the four categories of entities in FastCat Team and one for all
the vocabularies.

The derived semantic network contains more than 18.5M RDF triples and is
currently exploited by the data exploration application (ResearchSpace; more
below) for supporting historians in finding answers to their information
needs. The full RDF datasets are publicly
available191919https://zenodo.org/record/6460841. The network contains
interconnected information for thousands of sailors, ships, locations,
organisations, voyages, and many other relevant activities, as well as
connections with publicly available resources (Geonames, Getty TGN).

Semantic network exploration. For enabling historians of SeaLiT and other
interested parties to explore the integrated data and find answers to their
information needs, we make use of ResearchSpace (Oldman and Tanase,, 2018).
ResearchSpace is a configurable, open source platform which operates over a
semantic network accessible through an RDF triplestore. It offers a variety of
functionalities, including a query building interface that supports users in
gradually building and running complex queries through a user-friendly
interface. The results can then be browsed and analysed quantitatively through
different visualisations, such as bar charts.

The platform was configured for the case of SeaLiT data, offering three main
data exploration functionalities: a) keyword search, b) semantic search
(through its assistive query building interface), and c) entities browsing
(per type of archival source). Fig. 5 shows a screen dump of the semantic
search functionality. The user inspects the “construction location of ships
that were constructed between 1830 and 1840”. The user first searched for
ships constructed between 1830 and 1840 (Fig.5-A), and then selected to group
the retrieved ships by their construction location (Fig.5-B) and visualise the
results in a bar chart (Fig.5-C). This query corresponds to a real information
need as provided by the historians of SeaLiT, and the answer is shown to the
user instantly (in less than one second). If the construction location is
unknown for a ship, this missing information is displayed in the chart (see
‘Unknown’ bar, Fig.5-D). The user can also start browsing information about
the retrieved ships (e.g. inspecting the owners of a ship and then other ships
owned by the same owner), visit the FastCat transcripts that provide the
corresponding information (for validation, or inspection of additional
contextual information), or download the results in CSV format for further
(external) analysis.

A deployment of the application is publicly
accessible.202020http://rs.sealitproject.eu/

Figure 5: Semantic search and results visualisation in ResearchSpace.

##  7 Quality Aspects and Lessons Learned

We discuss data quality aspects as well as relevant lessons learned from the
application of the proposed workflow model in maritime history research.

###  7.1 Quality Aspects

Every workflow cycle ends up with semantic data that in some cases may suffer
low quality characteristics, making the data practically difficult to be
exploited for the needs of research. In literature, data quality is commonly
considered as “fitness for use” as well as an indicator of data usability
(Pipino et al.,, 2002; Wang and Strong,, 1996), and several dimensions and
metrics for measuring data quality have been proposed (Pipino et al.,, 2002;
Zaveri et al.,, 2016). Although studying quality factors in detail is out of
the scope of this paper, below we focus on three main quality dimensions of
the semantic data that can significantly affect the quantitative analysis
process: completeness, consistency, conciseness.

Data completeness. A quality dimension that can be easily assessed in the
context of a schema/ontology or the particular use case scenario (Zaveri et
al.,, 2016). The lack of essential information, like missing dates and
locations of events, or names and professions of actors of a registry, may
affect the research analysis and the evidence for making a decision about a
historical subject.

Data consistency. This dimension can be viewed from a number of perspectives
(Zaveri et al.,, 2016; Hassenstein and Vanella,, 2022). Our perspective
comprises the schema-based and the value-based (or representational)
consistency. Schema-based consistency can be evaluated against a particular
schema/ontology. It prevents modeling issues, like the incompatible
attribution/interlinking of the entities, and averts potential reasoning
malfunction. For example, assigning ‘tonnage’ to a person (instead of a ship)
makes no sense, and under particular reasoning premises it may produce
inaccurate inference that people were used for the transportation of goods.
Value-based consistency concerns the format and the structure of comparative
values (numbers, dates, measurement values) to enable comparability.
Magnitudes, dimensions, quantities, time-spans, dates, places’ coordinates,
etc., to be effectively compared, they have to align their reference points or
units of measurement.

Data conciseness. This quality dimension comprises two perspectives: schema-
level conciseness and instance-level conciseness (Zaveri et al.,, 2016; Mendes
et al.,, 2012). Schema-level conciseness means that the data does not contain
equivalent attributes with different names (responsibility of the data
modeling engineer), while instance-level conciseness means that the data does
not contain equivalent objects with different identifiers (highly-dependant on
the quality of the curation process).

###  7.2 Lessons Learned

Next we present issues related to data quality that we faced while
implementing the workflow and which should be taken into account.

Missing information.  Missing values are very common and an important-to-know
information for researchers because they can affect the accuracy of
quantitative (statistical) analysis. This is related to the completeness
quality aspect described above. When a piece of information is not provided in
the original source, the corresponding cell in the data entry system is left
empty. The data exploration system must consider such empty values while
aggregating and showing information.

Data entry errors. Errors in the transcripts during data entry are common,
such as accidentally filling the wrong column in a table, or putting the
information in the wrong place due to misunderstanding. This is related to the
schema-based consistency quality aspect described above. Such errors are
directly reflected in the data exploration interfaces and can spoil user
experience. Thus, it is important to allow researchers visit the original
transcripts for validation or making corrections. Moreover, offering
mechanisms in the user interface that support users to avoid such errors
during data entry can limit the problem.

Non-consistent comparative values. It is very common that comparative values,
such as dates, dimensions, quantities, location coordinates, are not
consistent across archival sources of different types, because of different
reference points or units of measurement, making difficult their use in
comparisons, filtering, etc. This is related to the value-based consistency
quality aspect described above. An additional (automated, semi-automated or
manual) step is needed for aligning such values, however without changing the
values as they appear in the original source. This can happen either during
data curation or during data transformation.

Costly data curation.  Low-quality data curation can reduce user satisfaction
and produce invalid analysis results. This is related to the instance-level
conciseness quality aspect described above. The cost of manual data curation
is relative to the size of the data that need curation (number of entity
instances, number of vocabulary terms). The process can be very time consuming
for researchers in cases such as SeaLiT where the number of entities and
vocabularies is high. Thus, there is a need for tools that automate as much as
possible curation without significantly affecting quality, e.g. through semi-
automatic processes, supervised algorithms, or application-specific machine
learning.

##  8 Conclusion

We presented a workflow model for holistic data management in archival
research: from transcribing and documenting a set of archival documents, to
curating the transcribed data, integrating it to a rich semantic network, and
then exploring and analysing the integrated data. The merits of the approach
is that it speeds up data entry, it is provenance-aware decoupling data entry
from data curation and integration, it is interactive as well as appropriate
for semantic interoperability, aiming at the production of sustainable data of
high value and long-term validity.

We have showcased the feasibility and effectiveness of the model in maritime
history research, and we have reported empirical results from its application
(about thirty users, twenty types of archival documents, more than 600
records, more than fifty vocabularies, more than 110,000 entity instances,
more than 18.5 million triples of integrated information).

Issues that are worth further research include: (a) semi-automated methods to
speedup data curation, (b) investigate the evolution requirements of the
semantic network, as proposed by Marketakis et al., (2021), (c) methods and
interfaces to support researchers in defining and updating the source schemas
by themselves.

### Acknowledgements

This work has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No
890861 (Project ReKnow), and ii) the European Research Council (ERC) grant
agreement No 714437 (Project SeaLiT).

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Optimizing in $\lambda$ gives for some constant $C>0$ and any $\delta>0$ the
following estimate

$\mathbb{E}_{X,N}\left[\chi\left(\frac{\sqrt{{2\pi}\,\psi_{\rho}(X)}N}{\delta}\right)\right]\leq
C\delta^{\frac{2\gamma}{2\gamma+1}}.$ (39)

Gathering the bounds (38) and (37) gives the bound

$\mathbb{P}\text{-a.s.},\,\exists
C(\omega)>0,\,\text{s.t.}\,\,\forall\delta>0,\,\,\mathbb{P}_{X}\left(\left|f_{n}(X)\right|\leq\delta\right)\leq
C\delta^{\frac{2\gamma}{2\gamma+1}}+\frac{C(\omega)}{\delta^{3}n^{\frac{4\theta}{5}}}.$
(40)

The rest of the proof follows the same strategy as in the proof of Theorem
1.3. Namely we write, for some $\lambda>0$ to be determined later

$\displaystyle\mathbb{E}_{X}\left[\mathcal{N}(g_{n},[0,2\pi])^{1+\eta/2}\right]$
$\displaystyle=$
$\displaystyle(1+\eta/2)\int_{0}^{+\infty}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds$
(41) $\displaystyle=$
$\displaystyle(1+\eta/2)\int_{0}^{n^{\lambda}}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds$
$\displaystyle+$
$\displaystyle(1+\eta/2)\int_{n^{\lambda}}^{\infty}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds.$

Then we write, using again repeatedly the Rolle Theorem

$\displaystyle\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}_{X}\left(|g_{n}(0)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}}{\left\lfloor s\right\rfloor!}\|g_{n}^{(\left\lfloor
s\right\rfloor)}\|_{\infty}\right)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}_{X}\left(|g_{n}(0)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}}{\left\lfloor s\right\rfloor!}M\right)$
$\displaystyle+\mathbb{P}_{X}\left(\|g_{n}^{(\left\lfloor
s\right\rfloor)}\|_{\infty}\geq M\right).$

Besides, via Sobolev estimates, and using the orthogonality of
$(\cos(kX),\sin(kX))$ with respect to the Lebesgue measure on $[0,2\pi]$ we
obtain that

$\displaystyle\mathbb{P}_{X}\left(\|g_{n}^{(\left\lfloor
s\right\rfloor)}\|_{\infty}\geq M\right)$ $\displaystyle\leq$
$\displaystyle\frac{1}{M^{2}}\left(\mathbb{E}_{X}\left[g_{n}^{\lfloor
s\rfloor}(X)^{2}\right]+\mathbb{E}_{X}\left[g_{n}^{\lfloor
s+1\rfloor}(X)^{2}\right]\right)$ $\displaystyle=$
$\displaystyle\frac{1}{M^{2}}~{}~{}\frac{1}{n}\sum_{k=1}^{n}\left(\left(\frac{k}{n}\right)^{2\lfloor
s\rfloor}+\left(\frac{k}{n}\right)^{2\lfloor
s+1\rfloor}\right)\left(\frac{a_{k}^{2}+b_{k}^{2}}{2}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C(\omega)}{M^{2}}.$

where the last inequality is again due to Birkhoff–Khinchine Theorem.
Together, the two previous bounds imply that

$\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)\leq\frac{C(\omega)}{M^{2}}+\mathbb{P}_{X}\left(|g_{n}(0)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}}{\left\lfloor s\right\rfloor!}M\right)\\\ $ (42)

Now, we can choose $M(s)=(1+|s|)^{\eta/2+1}$ and notice that, for $s$ large
enough, $\frac{(2\pi)^{\left\lfloor s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}<1$. As a result, provided that $s$ is large enough, we get by
Markov inequality and the fact that $x\mapsto|\log(x)|$ decreases on $]0,1[$:

$\displaystyle\mathbb{P}_{X}\left(|g_{n}(0)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{X}\left(\left|\log|f_{n}(X)|\right|\geq\left|\log\left(\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{\mathbb{E}_{X}\left[\left|\log(|f_{n}(X)|)\right|^{1+\eta}\right]}{\left|\log\left(\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)\right|^{1+\eta}}.$

Now, we rely on Lemma 4.4 which guarantees that for some constant
$C(\omega)>0$ and $\epsilon>0$ small enough we have

$\forall n\geq 1,\,\mathbb{E}_{X}|\log(|f_{n}(X)|)|^{1+\eta}\leq
C(\omega,\epsilon)n^{\epsilon}.$

Hence, for all $\eta>0$ and $s$ large enough, (and recalling that
$g_{n}(0)=f_{n}(X)$) we get

$\sup_{n\geq 1}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}\right)\leq\frac{C(\omega,\eta)n^{\epsilon}}{s^{1+\eta}}.$
(43)

Plugging the estimate (43) inside (42) ensures that, for $s$ large enough,
$\mathbb{P}$-a.s.,

$\forall n\geq
1,\,\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)\leq\frac{C(\omega)}{(1+s)^{2+\eta}}+\frac{C(\omega,\eta)n^{\epsilon}}{s^{1+\eta}}.$
(44)

Now we combine (44) and (41) and we obtain

$\displaystyle\mathbb{E}_{X}\left[\mathcal{N}(g_{n},[0,2\pi])^{1+\eta/2}\right]$
$\displaystyle=$
$\displaystyle(1+\eta/2)\int_{0}^{+\infty}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds$
$\displaystyle=$
$\displaystyle(1+\eta/2)\int_{0}^{n^{\lambda}}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds$
$\displaystyle+$
$\displaystyle(1+\eta/2)\int_{n^{\lambda}}^{\infty}s^{\eta/2}\mathbb{P}_{X}\left(\mathcal{N}(g_{n},[0,2\pi])>s\right)ds$
$\displaystyle\leq$
$\displaystyle(1+\eta/2)\int_{0}^{n^{\lambda}}s^{\eta/2}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)ds$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq-N-PX}}}{{+}}$
$\displaystyle(1+\eta/2)\int_{n^{\lambda}}^{\infty}s^{\eta/2}\left(\frac{C(\omega)}{(1+s)^{2+\eta}}+\frac{C(\omega,\eta)n^{\epsilon}}{s^{1+\eta}}\right)ds.$

Then, in view of using the estimate (40) we set $a_{n}$ the least positive
number such that

$s>a_{n}\Rightarrow\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}\leq\frac{1}{n^{\frac{\theta}{5}}}.$

Due to the growth of the term $\lfloor s\rfloor!$ it is clear that for $n$
large enough we have $a_{n}<n^{\lambda}$.

$\begin{array}[]{lll}&I:=\displaystyle{\int_{0}^{n^{\lambda}}s^{\eta/2}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)ds}\\\
&=\displaystyle{\int_{0}^{a_{n}}s^{\eta/2}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}\right)ds+\int_{a_{n}}^{n^{\lambda}}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor s\right\rfloor!}\right)}\\\
&\leq\displaystyle{\int_{0}^{a_{n}}s^{\eta/2}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}\right)ds+n^{\lambda}\mathbb{P}_{X}\left(|f_{n}(X)|\leq\frac{1}{n^{\frac{\theta}{5}}}\right)}\\\
&\displaystyle{\stackrel{{\scriptstyle\eqref{a.s.-esti8}~{}\text{with}~{}\delta=n^{-\frac{\theta}{5}}}}{{\leq}}}\displaystyle{C(\omega)\left(\left(\frac{1}{n^{\frac{\theta}{5}}}\right)^{\frac{2\gamma}{2\gamma+1}}+\frac{1}{n^{\frac{\theta}{5}}}\right.+\left.\int_{0}^{a_{n}}\left(\frac{(2\pi)^{\left\lfloor
s\right\rfloor}M(s)}{\left\lfloor
s\right\rfloor!}\right)^{\frac{2\gamma}{2\gamma+1}}ds+n^{\lambda}\times\frac{1}{n^{\frac{\theta}{5}}}\right)}.\end{array}$

Provided that $\lambda$ is chosen small enough, each of the above term is
uniformly bounded in $n$ which guarantees the desired uniform integrability
and achieves the proof.

######  Remark 4.1.

In the same way, a similar proof would give that $\mathbb{P}$-almost surely,
for any compact $[a,b]$ of $[0,2\pi]$,

$\lim_{n\to+\infty}\frac{\mathcal{N}(f_{n},[a,b])}{n}=\frac{b-a}{\pi\sqrt{3}}.$

Indeed, assuming this time that $X$ is uniformly distributed over $[a,b]$, one
may study the convergence of the stochastic process
$g_{n}(x)=f_{n}\left(X+\frac{x}{n}\right)$ towards a non-degenerate limit and
apply the same strategy.

##  5 Appendix

###  5.1 Birkhoff–Khinchine Theorem for Gaussian sequences

In this section, we recall the necessary material about Gaussian ergodicity
and the technical details ensuring the validity of Lemma 3.4 and 3.9 stated in
Sections 3.2 and 3.3 respectively. First, one can build the stationary
sequence $\\{a_{k}\\}_{k\geq 1}$ as the coordinates on the space
$\mathbb{R}^{\mathbb{N}}$ endowed with the cylindrical topology and equipped
with the Gaussian measure $m$ defined as

$\forall p\geq
1,\forall(A_{1},\cdots,A_{p})\in\mathcal{B}\left(\mathbb{R}\right)^{p},\,m\left(A_{1}\times
A_{2}\times\cdots\times
A_{p}\times\mathbb{R}\times\mathbb{R}\times\cdots\right)=\mathbb{P}\left(\bigcap_{i=1}^{p}\\{a_{i}\in
A_{i}\\}\right).$

Then, the shift operates on this space and preserves the measure $m$ since the
sequence $\\{a_{k}\\}_{k\geq 1}$ is stationary. We refer to [CFS82, pages 188]
for an introduction to these kinds of dynamical systems. One can use the
celebrated Birkhoff–Khinchine Theorem for this transformation, see e.g.
[CFS82, pages 11], and we get that

$\mathbb{P}-\text{a.s.},\;\;\frac{1}{n}\sum_{k=1}^{n}a_{k}^{2}\to\mathbb{E}\left[(x_{1},\cdots,)\mapsto
x_{1}^{2}\,\Big{|}\,\mathcal{I}\right],$

where $\mathcal{I}$ is the sigma field generated by all the mappings from
$\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$ that are shift-invariant $m$-almost
surely. In particular, $\mathbb{P}-$almost surely,
$\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}a_{k}^{2}+b_{k}^{2}$ has a finite limit as
$n$ goes to infinity, which is the key ingredient to establish Lemma 3.4.

###### Proof of Lemma 3.4.

Recall that $m$ is a large positive integer and $n$ is defined as the unique
integer such that $n^{7}<m\leq(n+1)^{7}$. By Cauchy–Schwarz inequality, we
have

$\left|\,\mathbb{E}_{X}\left[e^{itf_{n^{7}}(X)}\right]-\mathbb{E}_{X}\left[e^{itf_{m}(X)}\right]\,\right|\leq
t\,\mathbb{E}_{X}\left[|f_{n^{7}}(X)-f_{m}(X)|^{2}\right]^{1/2}.$

Otherwise, we have the estimate

$\begin{array}[]{ll}&\displaystyle{\mathbb{E}_{X}\left[|f_{n^{7}}(X)-f_{m}(X)|^{2}\right]\leq
2\left(1-\sqrt{\frac{n^{7}}{m}}\right)^{2}\mathbb{E}_{X}\left[f_{n^{7}}(X)^{2}\right]}\\\
\\\
&+\displaystyle{2\,\mathbb{E}_{X}\left[\left|\frac{1}{\sqrt{m}}\sum_{k=n^{7}+1}^{m}a_{k}\cos(kX)+b_{k}\sin(kX)\right|^{2}\right]}\\\
\\\
&=\displaystyle{2\left(1-\sqrt{\frac{n^{7}}{m}}\right)^{2}\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}+\frac{2}{m}\sum_{k=n^{7}+1}^{m}\frac{a_{k}^{2}+b_{k}^{2}}{2}}.\end{array}$
(45)

By Birkhoff–Khinchine Theorem, we have first

$\left(1-\sqrt{\frac{n^{7}}{m}}\right)^{2}\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}\sim\frac{1}{4n^{9}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}=O\left(\frac{1}{n^{2}}\right).$

Besides, one may write

$\displaystyle\frac{1}{m}\sum_{k=n^{7}+1}^{m}\frac{a_{k}^{2}+b_{k}^{2}}{2}\leq\frac{1}{n^{7}}\sum_{k=n^{7}+1}^{(n+1)^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}$
$\displaystyle=\frac{(n+1)^{7}}{n^{7}}\frac{1}{(n+1)^{7}}\sum_{k=1}^{(n+1)^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}-\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}$
$\displaystyle=$
$\displaystyle\left(\underbrace{\frac{1}{(n+1)^{7}}\sum_{k=1}^{(n+1)^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}-\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}}_{:=R_{n}}\right)+\underbrace{\frac{\sum_{k=0}^{6}\binom{7}{k}n^{k}}{n^{7}}}_{=O\left(\frac{1}{n}\right)}\times\underbrace{\frac{1}{(n+1)^{7}}\sum_{k=1}^{(n+1)^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}}_{=O\left(1\right)}.$

Next we write

$\begin{array}[]{ll}R_{n}&=\displaystyle{\sum_{k=1}^{n^{7}}\left(\frac{a_{k}^{2}+b_{k}^{2}}{2}\right)\left(\frac{1}{(n+1)^{7}}-\frac{1}{n^{7}}\right)+\frac{a_{(n+1)^{7}}+b_{(n+1)^{7}}}{2(n+1)^{7}}}\\\
\\\
&=\displaystyle{\underbrace{\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\left(\frac{a_{k}^{2}+b_{k}^{2}}{2}\right)}_{=O\left(1\right)}\underbrace{\left(\frac{1}{(1+1/n)^{7}}-1\right)}_{=O\left(\frac{1}{n}\right)}+\frac{1}{n+1}~{}\underbrace{\frac{a^{2}_{(n+1)^{7}}+b^{2}_{(n+1)^{7}}}{2(n+1)^{6}}}_{=o(1)}}.\end{array}$

Let us finally detail the $o(1)$ in the above equation. For any sequence
$\\{X_{k}\\}_{k\geq 1}$ of standard Gaussian random variables, any $\beta>0$
and any $\epsilon>0$ we have

$\sum_{k=1}^{\infty}\mathbb{P}\left(|X_{k}|>\epsilon
k^{\beta}\right)\leq\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi}}\int_{\epsilon
k^{\beta}}^{\infty}e^{-\frac{x^{2}}{2}}dx<\infty.$

Then Borel–Cantelli Lemma implies that $X_{n}/n^{\beta}\to 0$ almost surely,
in particular in our case, we have $|a_{(n+1)^{7}}/(n+1)^{3}|=o(1)$, hence the
result.

∎

###### Proof of Lemma 3.9.

Let us rewrite

$Z_{n}(X,t,\lambda)=\sum_{p=1}^{M}\lambda_{j}g_{n}(t_{p})=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_{k}\alpha_{k,n}(X)+b_{k}\beta_{k,n}(X),$

where

$\alpha_{k,n}(X):=\sum_{p=1}^{M}\lambda_{p}\cos\left(kX+\frac{kt_{p}}{n}\right),\quad\beta_{k,n}(X):=\sum_{p=1}^{M}\lambda_{p}\sin\left(kX+\frac{kt_{p}}{n}\right).$

Due to the orthogonality of trigonometric functions, a straightforward
computation then yields the following orthogonality relations, for all $1\leq
k\leq n$ and $1\leq k^{\prime}\leq n^{\prime}$

$\mathbb{E}_{X}\left[\alpha_{k,n}(X)\alpha_{k^{\prime},n^{\prime}}(X)\right]=\mathbb{E}_{X}\left[\beta_{k,n}(X)\beta_{k^{\prime},n^{\prime}}(X)\right]=\delta_{k,k^{\prime}}\times\frac{1}{2}\sum_{p,q=1}^{M}\lambda_{p}\lambda_{q}\cos\left(\frac{kt_{p}}{n}-\frac{kt_{q}}{n^{\prime}}\right),$

and

$\mathbb{E}_{X}\left[\alpha_{k,n}(X)\beta_{k^{\prime},n^{\prime}}(X)\right]=\delta_{k,k^{\prime}}\times\frac{1}{2}\sum_{p,q=1}^{M}\lambda_{p}\lambda_{q}\sin\left(\frac{kt_{p}}{n}-\frac{kt_{q}}{n^{\prime}}\right),$

so that by symmetry

$\mathbb{E}_{X}\left[\alpha_{k,n}(X)\beta_{k^{\prime},n}(X)\right]=0.$

In particular, we have

$\mathbb{E}_{X}\left[\alpha_{k,n}(X)^{2}\right]=\mathbb{E}_{X}\left[\beta_{k,n}(X)^{2}\right]\leq\frac{1}{2}\times||\lambda||_{1}^{2}.$

and

$\left\\{\begin{array}[]{l}\displaystyle{\mathbb{E}_{X}\left[(\alpha_{k,n}(X)-\alpha_{k,n^{\prime}}(X))^{2}\right]\leq
2\pi||\lambda||_{1}^{2}\left|\frac{k}{n}-\frac{k}{n^{\prime}}\right|},\\\ \\\
\displaystyle{\mathbb{E}_{X}\left[(\beta_{k,n}(X)-\beta_{k,n^{\prime}}(X))^{2}\right]\leq
2\pi||\lambda||_{1}^{2}\left|\frac{k}{n}-\frac{k}{n^{\prime}}\right|}.\end{array}\right.$
(46)

Recall that $m$ is a positive integer and $n$ is defined as the unique integer
such that $n^{7}<m\leq(n+1)^{7}$. By triangular inequality and Cauchy–Schwarz
inequality, we have

$\begin{array}[]{rl}\left|\mathbb{E}_{X}\left[e^{iZ_{n^{7}}(X,t,\lambda)}\right]-\mathbb{E}_{X}\left[e^{iZ_{m}(X,t,\lambda)}\right]\right|&\leq\mathbb{E}_{X}\left[|Z_{n^{7}}(X,t,\lambda)-Z_{m}(X,t,\lambda)|\right]\\\
\\\
&\leq\sqrt{\mathbb{E}_{X}\left[U^{2}\right]}+\sqrt{\mathbb{E}_{X}\left[V^{2}\right]}+\sqrt{\mathbb{E}_{X}\left[W^{2}\right]}\end{array}$

where

$\begin{array}[]{ll}U&:=\displaystyle{\left(\frac{1}{\sqrt{n^{7}}}-\frac{1}{\sqrt{m}}\right)\left(\sum_{k=1}^{n^{7}}a_{k}\alpha_{k,n^{7}}(X)+b_{k}\beta_{k,n^{7}}(X)\right)},\\\
\\\
V&:=\displaystyle{\frac{1}{\sqrt{m}}\left(\sum_{k=1+n^{7}}^{m}a_{k}\alpha_{k,m}(X)+b_{k}\beta_{k,m}(X)\right)},\\\
\\\
W&:=\displaystyle{\frac{1}{\sqrt{m}}\left(\sum_{k=1}^{n^{7}}a_{k}\left(\alpha_{k,n^{7}}(X)-\alpha_{k,m}(X)\right)+b_{k}\left(\beta_{k,n^{7}}(X)-\beta_{k,m}(X)\right)\right)}.\end{array}$

Using the orthogonality relations above, and again Birkhoff–Khinchine Theorem,
we have then

$\mathbb{E}_{X}[U^{2}]\leq||\lambda||_{1}^{2}\left(\frac{1}{\sqrt{n^{7}}}-\frac{1}{\sqrt{m}}\right)^{2}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}=||\lambda||_{1}^{2}\underbrace{\left(1-\sqrt{\frac{n^{7}}{m}}\right)^{2}}_{O(1/n^{2})}\underbrace{\left(\frac{1}{n^{7}}\sum_{k=1}^{n^{7}}\frac{a_{k}^{2}+b_{k}^{2}}{2}\right)}_{O(1)}.$

In the same way and proceeding as in the proof of Lemma 3.4 above, we get

$\mathbb{E}_{X}[V^{2}]\leq||\lambda||_{1}^{2}\times\frac{1}{m}\sum_{k=1+n^{7}}^{m}\frac{a_{k}^{2}+b_{k}^{2}}{2}=O\left(\frac{1}{n}\right).$

Finally, using again the orthogonality relations and Equation (46), we obtain

$\mathbb{E}_{X}[W^{2}]\leq||\lambda||_{1}^{2}\times
4\pi\times\left(1-\frac{n^{7}}{m}\right)\times\left(\frac{1}{m}\sum_{k=1}^{n^{7}}(a_{k}^{2}+b_{k}^{2})\right)=O\left(\frac{1}{n}\right).$

As a conclusion, we get that

$\left|\mathbb{E}_{X}\left[e^{iZ_{n^{7}}(X,t,\lambda)}\right]-\mathbb{E}_{X}\left[e^{iZ_{m}(X,t,\lambda)}\right]\right|=O\left(\frac{1}{\sqrt{n}}\right)=O\left(\frac{1}{m^{1/14}}\right).$

hence the result. ∎

###  5.2 Trigonometric kernels and convolutions

Recall the definitions of the kernel $K_{n}^{t,\lambda}$ and its normalized
version $\bar{K}_{n}^{t,\lambda}$ given in Lemma 3.5.

$K_{n}^{t,\lambda}(x):=\frac{1}{n}\left|\sum_{p=1}^{M}\lambda_{p}e^{i\frac{(n+1)}{2n}t_{p}}\frac{\sin\left(\frac{n}{2}(x+\frac{t_{p}}{n})\right)}{\sin\left(\frac{x+\frac{t_{p}}{n}}{2}\right)}\right|^{2},\quad\bar{K}_{n}^{t,\lambda}(x):=\frac{{2\pi}\,K_{n}^{t,\lambda}(x)}{\int_{0}^{2\pi}K_{n}^{t,\lambda}(x)dx}.$

######  Lemma 5.1.

The function $\bar{K}_{n}^{t,\lambda}$ is a good trigonometric kernel, i.e. it
satisfies the following properties:

  1. 1.

$\bar{K}_{n}^{t,\lambda}\geq 0$ and
$\frac{1}{2\pi}\int_{0}^{2\pi}\bar{K}_{n}^{t,\lambda}(x)dx=1.$

  2. 2.

For $n$ large enough, there exists a constant $C=C(t,\lambda)>0$ such that
uniformly in $x\in[0,2\pi]$

$\sup_{x\in[0,2\pi]}\bar{K}_{n}^{t,\lambda}(x)\leq
Cn,\qquad\bar{K}_{n}^{t,\lambda}(x)\leq
C\left(\sum_{p=1}^{M}\frac{1}{n\left(x+\frac{t_{i}}{n}\right)^{2}}\right).$

  3. 3.

For any $\delta>0$ and for $n$ large enough,

$\lim_{n\to+\infty}\int_{|x|>\delta}\bar{K}_{n}^{t,\lambda}(x)dx=0.$

###### Proof.

The first point results from the fact that $K_{n}^{t,\lambda}$ is trivially
non negative and from the very definition of $\bar{K}_{n}^{t,\lambda}$. Notice
that by Lemma 3.7, as $n$ goes to infinity
$\int_{0}^{2\pi}K_{n}^{t,\lambda}(x)dx$ converges to

${2\pi}\sum_{p,q=1}^{M}\lambda_{p}\lambda_{q}\sin_{c}\left(t_{p}-t_{q}\right)=\mathbb{E}\left[\left|\sum_{p=1}^{M}\lambda_{p}N_{t_{p}}\right|^{2}\right]>0$

where $(N_{t})$ is the standard Gaussian $\sin_{c}$ process, which is known to
be non degenerate. Therefore, for $n$ large enough,
$\int_{0}^{2\pi}K_{n}^{t,\lambda}(x)dx$ is positive and we can give all the
estimates on $K_{n}^{t,\lambda}$ without loss of generality. Precisely, for
the second point, remark that by triangular inequality, for all
$x\in[0,2\pi]$,

$K_{n}^{t,\lambda}(x)\leq
2^{p-1}\sum_{i=1}^{p}\lambda_{i}^{2}K_{n}\left(x+\frac{t_{i}}{n}\right).$

where $K_{n}$ is the standard Fejér kernel. The desired estimates then follow
directly from the ones established in Lemma 2.1 for the standard Fejér kernel
$K_{n}$. The third point is an immediate consequence of the second, since for
$n$ large enough, $K_{n}^{t,\lambda}(x)=O(1/n)$ for $|x|>\delta$. ∎

We can now give the proof of Lemma 3.6, i.e. the Fejér–Lebesgue type
convergence associated to the kernel $\bar{K}_{n}^{t,\lambda}$.

###### Proof of Lemma 3.6.

The proof is very close from the one of Lemma 2.2, with the slight difference
that, contrary to $K_{n}$ and $L_{n}$, the new kernel
$\bar{K}_{n}^{t,\lambda}$ is not even. Let us set

$\varphi_{x}^{\pm}(t):=\mu_{\rho}([0,x\pm
t])-t\psi_{\rho}(x),\quad\Phi_{x}^{\pm}(t):=\int_{0}^{t}|d\varphi_{x}^{\pm}(u)|.$

Set in the same way $E^{\pm}:=\\{x\in[0,2\pi],\Phi_{x}^{\pm}(t)=o(t)\\}$. By
Theorem 8.4, p 106 of [Zyg02], Lebesgue almost all point $x$ in $[-\pi,\pi]$
belongs to the set $E^{+}\cap E^{-}$. Otherwise, we have the representation

$\bar{K}_{n}^{t,\lambda}\ast\mu_{\rho}(x)-\psi_{\rho}(x)=\frac{1}{{2\pi}}\left(\int_{0}^{\pi}\bar{K}_{n}^{t,\lambda}(u)d\varphi_{x}^{-}(u)+\int_{0}^{\pi}\bar{K}_{n}^{t,\lambda}(-u)d\varphi_{x}^{+}(u)du\right).$

The rest of the proof then follows the exact same lines as the one of Lemma
2.2. Indeed, for all $x\in E^{+}\cap E^{-}$, using the estimates for
$\bar{K}_{n}^{t,\lambda}$ of the second point of Lemma 5.1 above and an
integration by parts, one deduces that the two last integrals converge to zero
as $n$ goes to infinty. ∎

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