# Mobile Phone Metadata for Development

Damien C. Jacques  Louvain-la-Neuve, Belgium

###### Abstract

Mobile phones are now widely adopted by most of the world population. Each
time a call is made (or a SMS sent), a Call Detail Record (CDR) is generated
by the telecom companies for billing purpose. These metadata provide
information on when, how, from where and with whom we communicate.
Conceptually, they can be described as a geospatial, dynamic, weighted and
directed network. Applications of CDRs for development are numerous. They have
been used to model the spread of infectious diseases, study road traffic,
support electrification planning strategies or map socio-economic level of
population. While massive, CDRs are not statistically representative of the
whole population due to several sources of bias (market, usage, spatial and
temporal resolution). Furthermore, mobile phone metadata are held by telecom
companies. Consequently, their access is not necessarily straightforward and
can seriously hamper any operational application. Finally, a trade-off exists
between privacy and utility when using sensitive data like CDRs. New
initiatives such as Open Algorithm might help to deal with these fundamental
questions by allowing researchers to run algorithms on the data that remain
safely stored behind the firewall of the providers.

###### keywords:

Mobile Phone Data , Call Detail Record , D4D , Sustainable Development Goals ,
Data Revolution , Big Data

††journal: None

##  1 Introduction

Over the past two decades, access to telecommunication services has seen
exponential growth. From around 100 million in 1995, the number of mobile
cellular subscriptions has risen to 7.4 billion worldwide in 2016 – the
equivalent of the entire world population [1]. No technology has ever spread
faster around the world [2]. This growth was primarily driven by wireless
technologies and liberalization of telecommunication markets, along with new
financing and technology, which have enabled faster and less costly network
rollout.

However, this does not mean that every person in the world has subscribed to a
mobile service. Because many individuals own several handsets or have multiple
subscriber identity module (SIM) cards, the number of subscribers, estimated
to 4.7 billion worldwide, is substantially lower than the number of
subscriptions [3]. This is because the number of subscriptions tend to
exaggerate the mobile phone penetration rate111The mobile phone penetration
rate is the number of active mobile phone users per 100 people within a
specific population. in developed economies. On the other hand, in many
developing countries, mobile phone access is higher than subscription numbers
would suggest. Access is indeed fostered in countries where sharing mobile
phones is a common practice, especially within large households [4]. In a
world bank report, the practical impact of the difference between subscription
and household penetration222Portion of total households having access to
mobile phone within a specific population. is clearly explained [5]:

> “Take Senegal, where the subscription penetration was 57 per 100 people in
> 2009, but household penetration was estimated to be 30 points higher at 87.
> This larger household size can dramatically extend access to mobile phones,
> considering that on average nine persons are in each Senegalese household.“
> It results that “several low-income nations have higher mobile phone home
> penetration than some developed economies. For example, Senegal, along with
> some other low- and middle-income economies, has a higher proportion of
> homes with mobile phones than either Canada or the United States“.

The economic potential of the mobile phone is tremendous. 2015 has been a year
of continued growth in the mobile industry, with operator revenues exceeding
$1 trillion. The mobile ecosystem generated 4.2% of Gross World Product and
directly support 17 million jobs [3]. The mobile telephony is increasingly
recognized as an essential tool of development by improving the flow of
information and providing a platform for financial services [6]. The more
striking example lies in the mobile money service which is now widely
established and brings financial inclusion to previously unbanked and
underbanked populations across the developing world (1.9 billion people
globally). Mobile phones are also a key platform to bring internet access to
people across the globe, particularly in developing regions where fixed
broadband services are prohibitively expensive, and fixed-line infrastructure
is limited. At the end of 2015, 2.5 billion individuals from the developing
world were accessing the internet through mobile devices [3].

Modern mobile phones (smartphones) are now integrated computers with dozens of
embedded sensors, such as accelerometer, digital compass, gyroscope, GPS,
microphone, and camera, which enable the emergence of several research
applications based on personal sensing [7]. These are promising avenues that
are believed to revolutionize many sectors of the economy, but the penetration
of smartphones is still low in Africa because of their higher cost. For
example, only 19% of the Senegalese population reported owning a smartphone in
2015 [8].

On the other hand, even the most basic handset passively generates a vast
amount of metadata leaving behind a digital trace333Also called digital
shadow, digital footprint or data exhaust. of the activity of its user. These
metadata provide information on when, how, from where and with whom we
communicate [9]. At first, researchers realized the potential of such data by
uploading tracking software into consenting subjects’ phones through the
Reality Mining project of the MIT444Note that as early as 1999, it was already
demonstrated that OD matrix could be obtained from the localisation of mobile
phones [10]. But the potential of CDRs for computational social science was
discovered later. [11]. They later gained access to actual metadata directly
from mobile network providers, leading to larger-scale research and greater
analytical power (e.g., 12). Until then, more and more datasets were opened up
to the scientific community, and mobile phone metadata are now seen as a
typical example of empirical data used in network science [13]. The
applications are tremendous, particularly for studies related to mobility,
social network and socio-demographics of people [9, 14, 15]. Several
initiatives have emerged, such as the Data For Development (D4D) challenge
organized by Orange, that provided datasets to the research community for
projects related to development. In a recent survey carried out by the World
Bank, mobile phone data appeared at the top position, just before satellite
imagery, in the Big Data sources used in SDG-related projects [16].

The objective of this paper is to introduce mobile phone metadata, in
particular call data records used by the companies for billing purpose, and
their potential for the SDGs. We first present some elements of mobile network
infrastructure, a prerequisite to understand the characteristics of data
collected by a mobile network operator. We then describe the specific features
that make CDRs unique and how they can be used to help achieving the SDGs.
Finally, we discuss the statistical limitations of such data and the risks
associated with their use (in particular, data access and privacy).

##  2 Elements of Mobile Network Operator Infrastructure

This section has been written based on material developed in Swenson et al.
[17], Tiru [18], Ricciato et al. [19], Janecek et al. [20], Ricciato et al.
[21].

Almost all Mobile Network Operators (MNO) in the world use two main mobile
technologies – GSM (Global System for Mobiles) and CDMA (Code Division
Multiple Access)555CDMA may disappear in favor of GSM network due to the
spread of the fastest and high-quality Long Term Evolution (LTE) technology
that uses a similar technology as GSM.. The market share of subscribers using
CDMA worldwide is 15-25% (mostly in North-America and some Asian countries)
and 75-85% for GSM (the rest of the world). The main difference between these
technologies is the radio signaling technology. The practical implication is
that a mobile device is tied to a particular network within CDMA network,
while a Subscriber Identity Module (SIM) card is tied to a specific network
within GSM network. It is, therefore, easier to switch mobile devices within
GSM networks thanks to SIM card’s portability. GSM phones without CDMA support
cannot run within CDMA network (and conversely).

A GSM network is a radio network of individual cells, known as Base
Transceiver Station (BTS). The BTS is responsible for transmitting and
emitting radio communications between the network and the mobile devices which
on the ground can be identified by the antenna tower and equipment (Figure 1,
left). To improve the network efficiency, BTSs are hierarchically grouped in
Location Area (LA) and controlled by a Base Station Controller (BSC) (Figure 1
and 2). The BSC is responsible for handover procedures666i.e. BTS switch when
a call in progress moves from one base station to another. within a single LA
between one BTS to another. BSCs are controlled by Mobile Switching Centres
(MSCs) that accommodates the Visitor Location Register (VLR) – the registry
for holding the information about the LA in which the mobile devices is
located (Figure 2). Finally, MSCs report to the Network Management System
(NMS) where all administrative and central procedures reside. Usually, only
data that are transmitted from MSCs to NMS are stored for different purposes
while lower level data traffic are deleted. NMS accommodates different
registries and databases that are important for the network functioning, in
particular, billing databases (Call Data Records777Call Detail Records and
Call Data Records are used interchangeably in this paper.). Components of
Mobile Positioning System, the system used for pinpointing users’ location for
emergency and security services (e.g., E112 and E911 directives in EU and US),
also resides in NMS and MSCs.

Figure 1: The two main sources of location data collected by Mobile Network
Operators: the Base Transceiver Station (shown in green and picture on the
left) is stored in the Call Detail Records and the Location Area (shown in
red) is stored in the Visitor Location Register (VLR).

Figure 2: Simplified structure of a GSM network. MSC stands for Mobile
Switching Center, BSC for Base Station Controller, LA for Location Area, BTS
for Base Transceiver Station, VLR for Visitor Location Register and CDR for
Call Detail Records. Arrows indicate the propagation of the signal needed to
locate the recipient when a call is initiated.

Different MNOs may share some of the network equipment. For example, it is not
uncommon that different MNOs share the same BTS. There are also special types
of MNOs (virtual MNOs) that do not possess any network infrastructure, but
instead rent it from other MNOs. In such case usually the virtual MNOs do not
have access to all operational data.

Mobile phones that are switched on are either in idle or in active state. In
idle state a mobile phone is not allocated any radio resource, but it
constantly evaluate if it needs to switch to another cell with a better signal
strength (it listens but does not transmit). Thus, in idle state a mobile
phone receives passively, which implies that the network is unable to identify
cell changes of idle mobile phones, except when such a switch is explicitly
requested by the mobile phone. Cell switches within one LA (from one BTS to
another) are not reported, but cell shifts from one LA to another are. It
turns out that the LA of any switched-on mobile phone is known at any time by
the MNO.

A mobile phone remains most of the time in idle state and only become active
during a call or a data transaction. When a mobile phone user dials a number
to make a call, a call initiation request is sent to the MSC. The MSC
validates the request by checking the user’s identity and airtime balance in
the records of its database. If valid, a connexion is initiated with the third
party, and the MSC requests the base station to move the mobile phone to an
unused voice channel so that the call can begin. Once a call is in progress,
the MSC adjusts the power transmitted by the mobile phone as it moves in and
out of the coverage area of each base station. When a mobile phone with a call
in progress moves from one BTS to another, handover procedure are
automatically managed by the network.

In summary, the state of the mobile phone (idle or active) determines the
temporal and spatial accuracy of the user location within the network data
system.

##  3 Mobile Phone Metadata

###  3.1 Call Detail Records (CDR)

CDRs typically include BTS location information of the caller and recipient
(starting cell), as well as time stamp and call duration (Table 1). However,
the information stored in CDRs really depends on each MNO as no standardised
structure exists (e.g., whether long calls are chunked into multiple CDRs)
[22]. Contrary to what the term suggests, CDRs are also used for SMS and data
connection (sometimes stored in separate data files). CDR data contain cell-
level locations, but only for active mobile phone engaged in voice call, SMS
or data connections. Their use for billing purpose implies an archived and a
constant update of the data without changing nor deleting old records. This
means that long time series are easily accessible for data analytics.

Table 1: Sample of typical call data records. Caller | Callee | Outgoing | Incoming | Timestamp | Call duration
---|---|---|---|---|---
SIM | SIM | BTS | BTS |  | (sec)
0458685984 | 0488595496 | 12 | 365 | 2018-01-18 15:22:12 | 456
0458685984 | 0458685984 | 12 | 25 | 2018-01-18 22:24:12 | 35
0469875254 | 0498563201 | 879 | 567 | 2018-01-19 08:47:10 | 125
(…) | (…) | (…) | (…) | (…) | (…)

###  3.2 Visitor Location Register (VLR)

The VLR is a dynamic database supporting the operation of the Mobile Switching
Center (MSC). Principally, the VLR caches temporary data about the current LA
location of all mobiles, both active and idle. Due to the completeness of VLR
data, VLR records provide an instantaneous description of the location of all
mobile phones, at LA level. VLR data are highly dynamic because LA locations
are updated continuously. As a result, bulk VLR reading can only be conducted
in real-time in the background of network operations.

###  3.3 Passive Monitoring Systems

Some MNOs track signaling and traffic exchange (e.g. for handover) in the
network through passive monitoring systems. These systems aim to assess and
resolve network operation and troubleshooting [22]. Using the network data,
passive monitoring systems are able to locate every mobile phone with great
accuracy, both in terms of time and space. Locations at cell and LA level are
provided, both for active and for idle mobile phones.

###  3.4 Selecting a dataset

Selecting a data source for analysis is a trade-off between spatial and
temporal resolution, and data accessibility [21]. CDRs provide information at
the highest spatial resolution (cell level, see Figure 4) but are event-based.
Therefore, data are only available when the user makes a call (or send an
SMS/data). This can be a limitation for mobility studies that require regular
location update (see Statistical limitations section). On the other hand, the
mobile phone usage (number of calls, when and where, etc.) might provide
precious information on the user’s socio-demographic profile. CDRs are stored
offline and therefore easily accessible. Because of that, these are, by far,
the data most frequently used in the literature [9].

VLR provide data at the finest temporal resolution but at LA level. The
spatial resolution of LA is much lower than BTS. For instance, the Ile-de-
France region has almost 10,000 BTS grouped in only 32 LAs, each of which has
between 150 and 500 BTS [23].

Lastly, passive monitoring systems leveraging signaling data combine the best
of the two worlds. However, the systematic acquisition of such data are
generally based on proprietary system and not available to all operators. As a
result, very little research has been conducted using these data [20, 24].

Other data collected by MNO that are relevant for research applications
include airtime credit purchases and customer client profiles. Airtime credit
purchases are useful to predict socio-economic status of population [25, 26,
27] while customer client profile enrich mobile phone metadata with personnal
information such as gender and age. However, while mobile phone operators have
access to all the information filed by their customers and their operational
data, they generally limit the access to only a sample of what is available
depending on their own privacy policies and the regulation on privacy
protection of each country.

The following sections further detail the specific characteristics of CDRs.

##  4 Data Features

Conceptually, CDRs can be described as a geospatial, dynamic, weighted and
directed network (Figure 3 – A). In the following sections, each of these
dimensions is further developed.

###  4.1 Network Dimension

A network is defined by its nodes (or vertex) and links (or edges, ties). In
CDRs, nodes are SIM cards ($\sim$ mobile phones) and links are the calls (or
SMS) exchanged between two SIMs. Furthermore, the links are directed from the
caller to the callee and the duration of calls weights each tie (Figure 3 –
A). By neglecting the geospatial dimension and aggregating (e.g., by taking
the sum) the total call duration between each pair of users over a given
period, a static representation of the CDRs network can be obtained (Figure 3
– B). After temporal aggregation, the link weights of the network might also
represent the number of calls or SMS exchanged between two users.

According to graph theory, such network can be represented through its
adjacency matrix $A_{ij}$ (Eq. 1).

$A_{ij}=\begin{pmatrix}0&A_{1,2}&\cdots&A_{1,n}\\\ A_{2,1}&0&\cdots&A_{2,n}\\\
\vdots&\vdots&\ddots&\vdots\\\ A_{m,1}&A_{m,2}&\cdots&0\end{pmatrix}$ (1)

The values of $A_{ij}$ represent the weight of the links (call duration,
number of calls or number of SMS) between node $i$ and $j$. Because the
network is directed, the matrix is asymmetric and $A_{ij}\neq A_{ji}$. Note
that $A_{ii}=0$ because one user cannot call himself.

The topological structure (degree, density, connectivity, etc.) of CDRs
network inform on the characteristics of people’s social networks. Such
analysis lead to the emergence of a new field of research called computational
social science [28]. One interesting possibility offered by CDR network is the
detection of social communities by identifying group of users that interact
more with each other than with the rest of the population. This requires
efficient algorithm capable to handle large dataset (see for instance the
’Louvain method’ in 29).

Figure 3: Schematic representations of CDRs data. Letters (A-D) represent SIM
cards ($\sim$ individuals), numbers (1-8) represent antenna coverage
approximated by a Voronoï tesselation, and arrows represent call direction
(head) and duration (width). (A) geo-spatial, dynamic, directed weighted
network (here weights are call duration), (B) static, directed weighted
network (over $t$ to $t+2$ period), and (C) dynamic trajectories of SIM cards.

###  4.2 Geospatial Dimension

As previously explained, each call in CDRs is geolocated at the BTS level
(Figure 4 – A) so that a mobile phone user can be located within the coverage
of this BTS. One BTS generally accommodate more than one antenna (typically
three). Therefore, the location of the user can be more precisely defined
within the coverage of one antenna (Figure 4 – B). However, MNOs might only
provide CDRs at BTS scale to preserve anonymity (e.g., the D4D dataset). The
coverage of one particular antenna depends on its technical characteristics
(power, technology, etc.) and the BTS density (Figure 5). More antennas are
required in areas with more traffic which means that in urban and sub-urban
areas, cell areas typically span between hundreds of meters (micro-cells) and
a few kilometers of diameter, while sparsely populated areas are covered by
few macro-cells. Smaller cells (pico-cell and femto-cells) can also be
deployed in highly crowded areas, such as shopping malls, train stations, or
airports. The antenna density is still considerably lower in developing
countries than in developed areas (see for instance Figure 5). It is worth
mentioning that analyzing the characteristics of the signal exchanged between
the phones and the BTS allow to reduce spatial uncertainty by inferring the
user-antenna distance (using response-delay and strength) (Figure 4 – C).
Finally, triangulation can further increase the spatial accuracy of the
estimation of the user’s location (Figure 4 – D). Such procedure usually
requires authorization and approval from the user except for emergency
response such as E112 and E911 directives in EU and US.

Figure 4: Location at (A) the base station level, (B) the sector level, (C)
the sector level knowing signal characteristics, (D) triangulation.

Figure 5: BTS maps of Orange Sonatel in Senegal (top) and Orange Mobistar in
Belgium (bottom).

###  4.3 Temporal Dimension

The temporal resolution of CDRs depends on the activity of mobile phone users.
The best case is a user multiplying short event (SMS or short calls). It is
worth noting that a long call does not increase the temporal resolution as
most of the time, only the starting cell (where the call was initiated) is
recorded in CDRs. While it is a limitation for mobility analysis, the temporal
patterns of mobile phone usage is interesting to study the socio-economic
profiles of population [30, 31].

On the other hand, the temporal dimension of CDRs allows studying the dynamic
of social networks. Saramäki and Moro [14] showed the great value of this
approach for different scales of analysis (Figure 6). For instance, one can
explore the resilience of social ties for an individual or a community after a
shock such as a loss (at individual level) or a disaster (at community level).

Finally, one key element of CDRs is their potential for near real time
applications. As the data are collected on the fly, they are virtually
instantly available for analysis. This aspect has significant implications for
post-disaster monitoring and early warning systems.

Figure 6: Temporal features of network at different structural and spatial
scales. Figure from Saramäki and Moro [14] reproduced with permission of the
authors.

###  4.4 Spatio-Temporal Dimension

The combination of both spatial and temporal dimension provide valuable
information for dynamic population mapping and mobility analysis as well as
for land use classification.

The most straightforward application of CDRs is the estimation of the number
of people at specific place and time. This implies modeling the relationship
between the number of active mobile phone users with the actual population
over time and space. Using census as calibration data, accurate estimations of
population can be obtained during night-time [32]. However, the fact that the
relationship between active mobile phone users and the actual population is
not independent of time and space makes it harder to accurately extrapolate
these models during day-time. This would require calibration data for daytime
population which are rarely available.

From the CDRs, a spatial trajectory can be computed for each user and use in
mobility analyses (Figure 3 – C). The quality of the inference of such an
approach depends on the frequency of the user activity (e.g., in Figure 3 – C,
mobility of user D will be better predicted than user A). In particular,
mobile phone activity is known to be bursty [33]. Users tend to place most of
their calls in short bursts, followed by long periods with no call activity,
during which information about the user’s location is lacking888This
observation also applies to several other human activities such as web
browsing, stock trading or library visits [34, 35].. However, despite their
temporal sparseness and spatial coarseness, CDRs still offer great insights
into the movement patterns of individuals and communities [36]. Furthermore,
Song et al. [37] showed that human mobility was highly predictable, regardless
of the distance traveled, due to the regularity of our daily mobility. A
typical application of mobility analysis with CDRs is the computation of
origin-destination matrices at different temporal scales. It allows
identifying different mobility dynamics such as daily commuting and long-term
migration.

On the other hand, the temporal signature of BTS activity can be used to
define land use patterns. This method has been applied in urban areas to make
the distinction, between residential, business, industrial and leisure areas
among others [38].

##  5 Applications of Mobile Phone Metadata for Development

The use of CDRs for data for development is relatively new. The growing
interest in the field has been triggered by telecom companies opening large
datasets to the research community. The Data For Development (D4D) challenge,
launched by Orange in 2013, was the first release of an extensive CDR database
from an African country (Ivory Coast) to the international research community
[39]. It was also the first CDRs-related project to be labeled as
‘development’, and gained huge publicity after the United Nations, the World
Economic Forum and several high-profile academic institutions (including MIT
and Cambridge University) endorsed it [40]. The initiative was very successful
and resulted in dozens of innovative projects developed by research labs from
around the world. Encouraged by this success, a second D4D challenge, with
data from Senegal, was organized in 2014. Once again, it resulted in several
creative projects [41]. The pioneer work of the Big Data for Good team of
Telefonica999Telefonica is a Spanish multinational broadband and
telecommunications provider serving over 200 million users in Latin America,
Europe, and the United States. in the development of algorithms for inferring
socio-economic welfare from mobile phone use patterns should also be
acknowledged.

Hereafter, we briefly present five contrasted applications of CDRs analysis
for development in five different developing countries (Kenya, Haiti, Rwanda,
Ivory Coast, Senegal). The aim here is not to give an exhaustive review of all
the possible applications of CDRs but rather a quick overview of what can been
done. For a detailed and broad review of CDRs data analysis, the reader is
referred to the works of Blondel et al. [9], Naboulsi et al. [15] and Saramäki
and Moro [14].

###  5.1 Health

One of the most common (and successful) use of CDRs data for development is
for epidemiological studies of human infectious diseases. For instance,
Wesolowski et al. [42] used CDRs from Kenya to identify the dynamics of human
carriers that drive parasite importation between regions. They analyzed the
regional travel patterns of nearly 15 million individuals over the course of a
year and characterized the degree of connectivity among different areas in
Kenya. Using a simple transmission model and malaria infection prevalence
data, they were then able to map the importation routes that contribute the
most to malaria epidemiology on regional spatial scales.

###  5.2 Post-Disaster Management

Another application that greatly benefits from mobility data derived from CDRs
is crisis management following a disaster. Lu et al. [43] analyzed the
movements of nearly two million SIM card holders before and after the 2010
earthquake in Haiti, finding that one-fifth of Port-au-Prince’s residents left
the city by three weeks after the disaster. They also show that the trajectory
of people fleeing from regions hit by the earthquake was highly correlated to
their mobility patterns during normal times. Such findings suggested that
population movements during disasters are significantly more predictable than
previously thought and highly influenced by people’s social support
structures.

###  5.3 Poverty and Socio-Economics Level

Assessing socio-economics levels, in particular poverty prevalence, is another
recent development of mobile phone metadata. Blumenstock et al. [30] showed
how the individual’s past history of mobile phone use can inferred his/her
socio-economic status using records of billions of interactions on Rwanda’s
largest mobile phone network. They validated their approach with a phone
surveys of a geographically stratified random sample of 856 individual
subscribers and using a DHS composite wealth index at micro-region level. In
Pokhriyal and Jacques [44], we explore this topic further.

###  5.4 Transportation

Using CDR data from the first D4D for the city of Abidjan (Ivory Coast),
Berlingerio et al. [45] evaluated which new routes would best improve the
existing transit network to increase ridership and user satisfaction, both in
terms of reduced travel and wait time. Four new routes have been proposed by
the optimization system (called AllAboard), resulting in an expected reduction
of 10% city-wide travel times.

###  5.5 Energy

The first prize of the D4D Senegal challenge was awarded to a research project
which assessed the contribution of mobile phone data for the development of
bottom-up energy demand models in Senegal [46]. Specifically, the research
team introduced a framework that combines mobile phone data analysis (mobile
phone activity was used as a proxy of the energy consumption), socio-economic,
geo-referenced data analysis, and state-of-the-art energy infrastructure
engineering techniques to assess the techno-economic feasibility of different
centralized and decentralized electrification options for rural areas in a
developing country. The result was a country map of electrification
recommended option between (i) extensions of the existing medium voltage grid,
(ii) diesel engine-based community-level Microgrids, and (iii) individual
household-level solar photovoltaic systems.

##  6 Statistical Limitations

CDRs are a good example of Big Data source that can be diverted from their
primary purpose to approximate socio-economic variables and population
mobility. As they are not designed for this purpose, this means that an
unavoidable bias will always impact any application based on these data. If
not properly understood, this could lead to serious misinterpretation of the
results and ultimately, have harmful impacts in misleading policy-makers. This
section reviews some of the sources of inaccuracy inherent to mobile phone
metadata.

###  6.1 Technical issues

MNOs suffer occasional down-time during which data are not recorded (missing
data). Furthermore, cells can also be deactivated for maintenance or resource
optimization (e.g., during low activity period such as nighttime). On the
other hand, incorrect data can arise at different level of the data collection
due to encoding or other technical issues (e.g. duplicated records, records
with incorrect time values, etc.).

###  6.2 Selection bias

People generating CDR data have selected themselves as data generators through
their activity. This is called a ‘selection bias’. First, while the
penetration of mobile phone is very high in the developing world, some
sociodemographic groups (typically young children and senior people) are still
left out of the analysis when considering mobile phone metadata. The adoption
base in Africa has been more traditionally skewed towards a wealthier,
educated, urban and predominantly male population [4, 47]. Additionally, one
SIM card does not necessarily correspond to one person. Figure 7 illustrates
all the possible association schemes between SIM and persons. In developing
countries, this is frequent that someone owns different SIM cards to be able
to switch between mobile carrier’s network depending on promotional campaigns.
Phone sharing is also a common practice among the poorest. On the other hand,
data access is most often limited to only one provider in the country. This
could be problematic if the choice of a MNO is correlated with the socio-
economic profile of individuals.

Figure 7: Possible association schemes between SIM and persons. Figure adapted
from Ricciato et al. [19]. M2M stands for Machine to Machine communication.

However, depending on the application, the impact of ownership bias might not
be as strong as expected. For instance, Wesolowski et al. [48] show that
mobility estimates are surprisingly robust to the substantial biases in phone
ownership across different geographical and socio-economic groups using 1-year
data of 15 million individuals in Kenya.

###  6.3 Spatial bias

Lacking data on antenna power and orientation, their coverages are generally
approximated by means of a Voronoï tessellation (Figure 3 – A). It assumes
that mobile phones always connect to the closest antenna. However, dozens of
factors play a role in the decision of the system to assign a specific cell to
a phone (e.g., signal strength, atmospheric conditions, traffic overload,
maintenance schedule). It is, therefore, possible that a person at the same
location, making five successive phone calls, will connect to five different
antennas. Designed for business and not tracking, CDRs provide information
that helps companies manage their operations, not track phones. On the other
hand, as it has been already mentioned, the spatial resolution of CDRs depends
on the BTS density. It means that remote and unpopulated areas, where
populations at risk (such as poor and food insecure population) are generally
found, have lower spatial resolution than urban areas due to a lower antenna
density. Finally, CDR data are always limited to one country and due to
technical challenge, the cross-border movements are difficult to capture. This
is an important limitation for large scale epidemiological studies because the
spread of a disease does not stop at the border. Other similar data sources
capturing geographic digital footprints (e.g. tweets) may be used to overcome
this limitation [49].

##  7 Data Access

Because private companies hold the data, there is no guarantee of access. It
requires an agreement between researchers and the telecom company. The
companies might be reluctant to provide access due the threat to subscribers
privacy that can result in a loss of customers. Therefore, their interest to
open/sell datasets is somewhat limited. Yet, it is generally possible to get
access to a particular data set for testing/research purposes, but still far
harder (for legal or commercial reasons) to extend this access for production
purposes.

Research teams and institutions have learned the hard way that even in case of
emergency situations, being granted access to CDRs can still remain an
unsurmountable obstacle. When the Ebola epidemic broke out in Sierra Leone,
Liberia, and Guinea in 2014, a group of academics and international
development actors began to call for the use of aggregated location data from
mobile phone networks in order to facilitate the response effort [50]. After
dozens of conference calls over many months involving over fifty participating
organizations (including several UN agencies), permission was finally granted
by the relevant local authorities – except for Liberia. Despite having the
highest death toll of any country that experienced the Ebola epidemic, Liberia
never released CDRs, in part due to concern about their ability to enforce
privacy protection [51]. This experience raises fundamental questions on the
trade-off between privacy and utility and how it can be adjusted according to
the level of emergency of a situation.

In the aim to respond to these concerns, the mobile phone industry association
– the GSMA – has developed a ‘Mobile for Development Intelligence’ programme
to persuade mobile providers from developing countries to share data with
researchers, industries and development organisations. However, their focus is
primarily on commercial outputs as the goal of their open data portal – Mobile
for Development Products and Services Trackers – is worded as:

> “offer the industry access to high quality data to help improve business
> decision making, increase total investment from both the commercial mobile
> industry and the development sector as well as to accelerate economic,
> environmental and social impact from mobile solutions.“ [52]

##  8 Data Privacy

A lot of personal information can be extracted from CDRs. Using such data, it
is easy to know where people live and work as well as tracking most of their
movement [53, 54]. Their social network can be characterized allowing, for
instance, the examination of the evolution of relationships over time [55].
The way people use their phone is also a good indicator of their personality.
For instance, de Montjoye et al. [56] showed how CDRs could be used to infer
five main traits of personality: openness, conscientiousness, extraversion,
agreeableness, and neuroticism (a socio-psychological model known as OCEAN).
Based on facebook data, the same model was used by Cambridge Analytica to
micro-target of campaign material to US voters with the purpose of influencing
the 2016 presidential campaign [57, 58].

To protect people’s privacy, mobile phone data are always anonymized, i.e, all
personal data such as name, address, phone number, etc., are either removed
from the database or replaced by a randomly generated number to avoid
identification101010This process is known as pseudo-anonymization.. Data are
then provided to a third party after a non-disclosure agreement (NDA) was
signed with the MNO. The purpose of the agreement is to prevent CDRs to be
shared to another party, and to define the scope of research questions that
will be explored with the data. Both the anonymization procedure and the NDA
are supposed to preserve the safety of users privacy.

However, if individual patterns are unique enough, additional information can
be used to link the data back to an individual. Using fifteen months of human
mobility data derived from CDRs, [59] showed that four randomly chosen points
(i.e., four places where a user was at a specific time) are enough to uniquely
characterize 95% of the users, whereas two randomly chosen points still
uniquely characterize more than half of the users.

Data aggregation allows to further strengthen privacy. In the case of CDRs,
several approach can be used. First, users can be aggregated by BTS. With this
aggregation, it is no more possible to track one specific user and mobility
analyses cannot be performed anymore. On the other hand, it is still relevant
for dynamic population mapping as it only requires the number of a users at a
specific place and time. This is also still useful to study spatial network
based on antenna-to-antenna traffic. To keep mobility analysis feasible,
temporal and/or spatial aggregation can be used. However, decreasing the
resolution comes with a loss in data utility so that a trade-off exists
between privacy protection and the preservation of data value (Figure 8).
Furthermore, [59] showed that blurring the spatial and temporal resolution
does not significantly impact the number of points needed to re-identify a
user in the database. Finally, to preserve privacy, noise can also be added to
some variables of the database (e.g. random spatial reallocation of BTS).

Figure 8: Schematic representation of the trade-off between privacy and
utility of personal data. Full line is the actual relationship and the dotted
line shows the ideal relationship. The figure is adapted from an OPAL
presentation (www.opalproject.org).

On the other hand, in today’s digital world, increasing privacy is only useful
if it is done for all sources at the same time. This issue was defined as the
’secondhand smoke problem’ by Lane et al. [7]:

> “[…] the secondhand smoke problem of mobile sensing creates new privacy
> challenges, such as: • How can the privacy of third parties be effectively
> protected when other people wearing sensors are nearby? • How can mismatched
> privacy policies be managed when two different people are close enough to
> each other for their sensors to collect information from the other party?“

With the aim to solve some of these issues, the Open Algorithm (OPAL) project
was set up by Orange, MIT Media Lab, Data-Pop Alliance, the World Economic
Forum and the Imperial College London [60]. The idea behind OPAL is to bring
about a paradigm shift in mobile metadata analysis by moving the code to the
data rather than data to the code. It means that instead of providing data
directly to researchers through NDA, CDRs (or other sensitive data) remain
behind the firewall of each provider. Only certified algorithms, meeting
predetermined privacy standards, can be run on the data in this secure
environment and only aggregated results are shared with the user
(www.opalproject.org). It allows facilitating data access while preserving
business and individual privacy. In Pokhriyal and Jacques [44], we illustrated
a similar approach where two models of poverty mapping based on disparate data
sources can be combined without the need to share the raw data.

##  9 Conclusions

In this paper, the specific features of mobile phone metadata were discussed
with a focus on applications for development. The amount of information held
in these data is fantastic. Among other, they have been used to model the
spread of infectious diseases, study road traffic, support electrification
planning strategies or map the socio-economic level of population. While
massive, CDRs are not statistically representative of the whole population due
to several sources of bias. Furthermore, data access and privacy are
significant challenges that are not necessarily straightforward to resolve.

While the challenges exist, the potential of such data might exceed the
limitations. Compared to traditional data collected to compute official
statistics, they are cost-effective and can provide faster or even near real-
time insights. They might also be used to test concepts and define future
research questions. On the other hand, the Reality Mining project demonstrated
that observed behavior using mobile phone metadata strongly differs from what
was self-reported by the same individuals [11]. This suggests that the
subjectivity of the subjects’ perception produces a significant bias in
traditional surveys. The objectivity coming from their exogeneity is,
therefore, another strength of mobile phone metadata.

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# A generalisation of Pisier homogeneous Banach algebra

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

In 1979 Pisier proved remarkably that a sequence of independent and
identically distributed standard Gaussian random variables determines, via
random Fourier series, a homogeneous Banach algebra $\mathscr{P}$ strictly
contained in $C(\mathbb{T})$, the class of continuous functions on the unit
circle $\mathbb{T}$ and strictly containing the classical Wiener algebra
$\mathbb{A}(\mathbb{T})$, that is,
$\mathbb{A}(\mathbb{T})\subsetneqq\mathscr{P}\subsetneqq C(\mathbb{T}).$ This
improved some previous results obtained by Zafran in solving a long-standing
problem raised by Katznelson. In this paper we extend Pisier’s result by
showing that any probability measure on the unit circle defines a homogeneous
Banach algebra contained in $C(\mathbb{T})$. Thus Pisier algebra is not an
isolated object but rather an element in a large class of Pisier-type
algebras. We consider the case of spectral measures of stationary sequences of
Gaussian random variables and obtain a sufficient condition for the
boundedness of the random Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\,\xi_{n}\exp(2\pi int)$ in the general
setting of dependent random variables $(\xi_{n})$.

Key words: homogeneous Banach algebra, random Fourier series, Gaussian
processes.
AMS 2010 Classification: 46J10, 42A20, 60B15, 60G10.

##  1 Introduction

Pisier [8] constructed a homogeneous Banach algebra $\mathscr{P}$ of
continuous functions on the unit circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$
such that $\mathbb{A}(\mathbb{T})\subsetneqq\mathscr{P}\subsetneqq
C(\mathbb{T})$ where $C(\mathbb{T})$ is the Banach algebra of continuous
functions on $\mathbb{T}$ and $\mathbb{A}(\mathbb{T})$ is the classical Winer
algebra of continuous functions on $\mathbb{T}$ with absolutely convergent
Fourier series. He further obtained that all 1-Lipschitzian functions operate
on $\mathscr{P}$. To fully describe $\mathscr{P}$, we shall consider a fixed
probability space $(\Omega,\mathscr{F},\mathbb{P})$ and a Gaussian Hilbert
space $\mathscr{H}$, that is, a closed subspace of $L^{2}(\mathbb{P})$ whose
elements are all Gaussian random variables. Pisier algebra $\mathscr{P}$ is
determined by a sequence $Z=(Z_{n})_{n\in\mathbb{Z}}$ of independent standard
Gaussian random variables. We recall that a Gaussian process on $\mathbb{T}$
is simply a family $\\{X(t):t\in\mathbb{T}\\}$ of random variables in
$\mathscr{H}$. For $f\in C(\mathbb{T})$ and $t\in\mathbb{T}$, the random
Fourier series $\sum_{n\in\mathbb{Z}}\hat{f}(n)Z_{n}e^{2\pi nit}$ converges
obviously in $L^{2}(\mathbb{P})$ because
$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}<\infty.$ The class $\mathscr{P}$ is
defined as the class of functions $f\in C(\mathbb{T})$ such that the Gaussian
process $S(f,Z)=\\{S(f,Z,t):t\in\mathbb{T}\\}$ consisting of the $L^{2}$-sums
$S(f,Z,t)=\sum_{n\in\mathbb{Z}}\hat{f}(n)Z_{n}e^{2\pi nit}$ has bounded sample
paths almost surely. Equivalently $\mathscr{P}$ is the class of functions
$f\in C(\mathbb{T})$ such that the Fourier random series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)Z_{n}e^{2\pi nit}$ converges in
$C(\mathbb{T})$. Other equivalent characterisations of $\mathscr{P}$ can be
obtained from the classical Billard theorem (see Cohen and Cuny [2] and Kahane
[3, p 58]).

Given a Borel probability measure $\mu$ on $\mathbb{T}$ and a continuous
function $f\in C(\mathbb{T})$, we shall associate a Gaussian process
$X_{f}=\\{X_{f}(t):t\in\mathbb{T}\\}$ such that for all $t,s\in\mathbb{T}$,

$\mathbb{E}(|X_{f}(t)-X_{f}(s)|^{2})=\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}d\mu(u)$

and consider the class $\mathscr{P}(\mu)$ of those continuous functions $f$ on
$\mathbb{T}$ for which the process $X_{f}$ has bounded paths (almost surely)
and set

$\|f\|_{\mathscr{P}(\mu)}=\mathbb{E}\left(\sup_{s,t,\in\mathbb{T}}|X_{f}(t)-X_{f}(s)|\right)+\|f\|_{\infty}.$

Pisier algebra $\mathscr{P}$ is $\mathscr{P}(\mu)$ where $\mu$ is the Lebesgue
measure on $\mathbb{T}$. We shall obtain that $\mathscr{P}(\mu)$ is a
homogeneous Banach algebra on which all 1-Lipschitz functions operate and
further we shall obtain a sufficient condition for
$\mathscr{P}(\mu)\subset\mathscr{P}$.

As already indicated, Pisier algebra arose from the study of random Fourier
series of i.i.d Gaussian random variables. We shall consider throughout a
discrete-time complex Gaussian process $\xi=(\xi_{n})_{n\in\mathbb{Z}}$ in the
Gaussian Hilbert space $\mathscr{H}$ such that $\mathbb{E}(\xi_{n})=0$ and
$\mathbb{E}(|\xi_{n}|^{2})=1$ for all $n\in\mathbb{Z}$. We shall further
assume that for all $n,m\in\mathbb{Z}$,
$\mathbb{E}(\xi_{n}\overline{\xi_{m}})$ depends only on $n-m$ and
$\mathbb{E}(\xi_{n}\xi_{m})=0$. This implies that the process
$\xi=(\xi_{n})_{n\in\mathbb{Z}}$ is stationary. Clearly for any
$Z\in\mbox{span}(\xi)$, $Z$ is a complex Gaussian variable with distribution
$f(z)=\exp(-|z|^{2}/\sigma^{2})/(\pi\sigma^{2})$ where
$\sigma^{2}=\mathbb{E}(|Z|^{2})$. The covariance function of $\xi$ is the
function $\gamma:\mathbb{Z}\to\mathbb{C}$ defined by
$\gamma(n)=\mathbb{E}(\xi_{0}\overline{\xi_{n}}).$ Clearly
$\gamma(-n)=\overline{\gamma(n)}$ and $\gamma(0)=1$. Such sequence $(\xi_{n})$
can be obtained by taking a sequence $(\zeta_{n})$ of real standard Gaussian
random variables such that $\mathbb{E}(\zeta_{n}\zeta_{m})$ depends only on
$(n-m)$ and thereafter take $\xi_{n}=(\zeta_{n}+i\zeta_{n}^{\prime})/\sqrt{2}$
where $(\zeta_{n}^{\prime})$ is an independent copy of $(\zeta_{n})$. Since
the covariance function $\gamma$ is positive semidefinite, the classical
Bochner theorem implies that there exists a Borel probability measure $\mu$ on
the unit circle $\mathbb{T}$ such that

$\displaystyle\mathbb{E}(\xi_{n}\overline{\xi_{m}})=\gamma(n-m)=\int_{\mathbb{T}}e^{2\pi
i(n-m)t}d\mu(t),\,\,n,m\in\mathbb{Z}.$ (1)

The measure $\mu$ is the spectral measure of the process $\xi$. If $\mu$ is
absolutely continuous with respect to the Lebesgue measure on $\mathbb{T}$,
its density $\varphi$ is called the spectral density function of the process
$\xi$. In the corresponding Banach algebra $\mathscr{P}(\mu)$ (where $\mu$ is
the spectral measure of $\xi$), it is easy to see that the formal random
Fourier series $\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi int}$ is almost
surely the formal Fourier series of the function $t\to X_{f}(t)$. This is true
because the Gaussian sequence $(\hat{f}(n)\xi_{n})_{n\in\mathbb{Z}}$ has the
same distribution with the sequence of the Fourier coefficients
$(\hat{X_{f}}(n))_{n\in\mathbb{Z}}$ of $X_{f}$ given by

$\hat{X_{f}}(n)=\int_{\mathbb{T}}X_{f}(t)e^{-2\pi int}dt.$

If $\mu$ has a density $\varphi$ such that $\varphi\in L^{p}(du)$ for some
$1<p<\infty$, then for any $f\in\mathscr{P}(\mu)$ and $t\in\mathbb{T}$, the
formal Fourier random series $\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi
int}$ converges in $L^{2}(\mathbb{P})$. We shall obtain a sufficient condition
on the covariance matrix of the sequence $(\xi_{n})$ for the almost surely
boundedness of the random Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi nit}$ that generalises a
classical condition only known for i.i.d variables. This is in particular
applied to the classical fractional Gaussian noise.

The study of random Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}\exp(2\pi int)$ where the random
variables $(\xi_{n})$ are interdependent is important as it builds a bridge
between the classical random series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)Z_{n}\exp(2\pi int)$ of independent variables
$(Z_{n})$ and the deterministic Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\exp(2\pi int)$ which is the limit case where
all the random variables $(\xi_{n})$ are the same. For this limit case the
problem of $L^{2}$-convergence becomes the problem of characterisation of the
classical class $\mathscr{C}$ of functions $f\in C(\mathbb{T})$ with
everywhere convergent Fourier series. (This is very complex since
$\mathscr{C}$ is beyond the Borel hierarchy in $C(\mathbb{T})$ (see Ajtai and
Kechris [1]).) This paper can be seen as a step toward the study of random
Fourier series in the Banach space $C(\mathbb{T})$ in the presence of
interdependent random variables which constitutes an important open problem.

##  2 Banach algebras defined by probability measures

Given a Borel probability measure $\mu$ on $\mathbb{T}$ and a $f\in
C(\mathbb{T})$, we can associate to $f$ the pseudo-distance $d_{f}$ on
$\mathbb{T}$ defined by

$d_{f}(t,s)=\left(\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}d\mu(u)\right)^{1/2},\,\,t,s\in\mathbb{T}.$

Since the space $L^{2}(\mu,\mathbb{C})$ embeds in a Gaussian Hilbert space,
there exists a Gaussian process $X_{f}=\\{X_{f}(t):t\in\mathbb{T}\\}$ such
that for all $t,s\in\mathbb{T}$,

$\mathbb{E}\left(X_{f}(t)\overline{X_{f}(s)}\right)=\int_{\mathbb{T}}f(t+u)\overline{f(s+u)}d\mu(u)$

and hence

$\mathbb{E}(|X_{f}(t)-X_{f}(s)|^{2})=d_{f}^{2}(t,s)\,\,\,\mbox{ and
}\mathbb{E}(|X_{f}(t)|^{2})=\int_{\mathbb{T}}|f(t+u)|^{2}d\mu(u).$

More precisely, one may consider an orthonormal basis $(e_{n})_{n\in A}$ of
$L^{2}(\mu,\mathbb{C})$ (where $A$ is a subset of $\mathbb{Z}$), a sequence of
independent and identically distributed (i.i.d) standard complex Gaussian
random variables $(Z_{n})$ and set for all $t\in\mathbb{T}$,

$X_{f}(t)=\sum_{n\in A}\langle f_{t},e_{n}\rangle Z_{n}$

where $f_{t}$ is the function defined on $\mathbb{T}$ by $f_{t}(u)=f(t+u)$ and

$\langle
g,e_{n}\rangle=\int_{\mathbb{T}}g(u)\overline{e_{n}(u)}d\mu(u),\,\,g\in
L^{2}(\mu,\mathbb{C}).$

(The convergence of the series $\sum_{n\in A}\langle f_{t},e_{n}\rangle Z_{n}$
in $L^{2}(\mathbb{P})$ is guaranteed by the fact that $(Z_{n})$ are
independent $N(0,1)$ variables.) Clearly the process $X_{f}$ depends linearly
on $f$. Let $\mathscr{P}(\mu)$ be the class of continuous functions $f$ on
$\mathbb{T}$ such that the process $X_{f}$ has almost surely bounded paths and
set

$\|f\|_{\mathscr{P}(\mu)}=\mathbb{E}\left(\sup_{s,t,\in\mathbb{T}}|X_{f}(t)-X_{f}(s)|\right)+\|f\|_{\infty}.$

The classical Pisier algebra $\mathscr{P}$ is $\mathscr{P}(\mu)$ where $\mu$
is the Lebesgue measure on $\mathbb{T}$. We have the following result.

###### Theorem 1

For every Borel probability measure $\mu$, the class $\mathscr{P}(\mu)$ is a
homogeneous Banach algebra on which all 1-Lipschitz functions operate.

  * Proof

The proof is a replication of Pisier’s original argument in [8] and the
following classical comparison principle: Let $X=\\{X(t):t\in\mathbb{T}\\}$
and $Y=\\{Y(t):t\in\mathbb{T}\\}$ be two real Gaussian processes. If

$\mathbb{E}|Y(t)-Y(s)|^{2}\leq\mathbb{E}|X(t)-X(s)|^{2}\,\,\,\,\mbox{ for all
}t,s\in\mathbb{T},$

then

$\mathbb{E}\left(\sup_{s,t\in\mathbb{T}}|Y(t)-Y(s)|\right)\leq\mathbb{E}\left(\sup_{s,t\in\mathbb{T}}|X(t)-X(s)|\right).$

In particular if $X$ has bounded paths almost surely then $Y$ has also bounded
paths almost surely. Here

$\mathbb{E}\left(\sup\\{|X(t)|:t\in\mathbb{T}\\}\right)=\sup\left(\left\\{\mathbb{E}\sup\\{|X(t)|:t\in
F\\}:F\mbox{ finite subset of }\mathbb{T}\right\\}\right).$

The comparison principle extends easily to complex Gaussian processes. Indeed,
write $X_{1}=\Re(X)$, $X_{2}=\Im(X)$ (both $X_{1}$ and $X_{2}$ are real
Gaussian processes) and set $X^{\prime}=X_{1}+X_{2}^{\prime}$ where
$X_{2}^{\prime}$ is a real Gaussian process which is an independent copy of
$X_{2}$. Then clearly

$\mathbb{E}\left(\sup_{t,s}|X(t)-X(s)|\right)\leq
2\mathbb{E}\left(\sup_{t,s}|X^{\prime}(t)-X^{\prime}(s)|\right),$
$\mathbb{E}\left(\sup_{t,s}|X^{\prime}(t)-X^{\prime}(s)|\right)\leq
2\mathbb{E}\left(\sup_{t,s}|X(t)-X(s)|\right)$

and

$\mathbb{E}|X^{\prime}(t)-X^{\prime}(s)|^{2}=\mathbb{E}|X_{1}(t)-X_{1}(s)|^{2}+\mathbb{E}|X_{2}^{\prime}(t)-X_{2}^{\prime}(s)|^{2}=\mathbb{E}|X(t)-X(s)|^{2}.$

Similarly for $Y$, define $Y_{1},Y_{2},Y_{2}^{\prime}$ and write
$Y^{\prime}=Y_{1}+Y_{2}^{\prime}$ so that
$\mathbb{E}|Y^{\prime}(t)-Y^{\prime}(s)|^{2}=\mathbb{E}|Y(t)-Y(s)|^{2}$. Hence
$\mathbb{E}|Y^{\prime}(t)-Y^{\prime}(s)|^{2}\leq\mathbb{E}|X^{\prime}(t)-X^{\prime}(s)|^{2}$
which implies

$\mathbb{E}\left(\sup_{t,s}|Y^{\prime}(t)-Y^{\prime}(s)|\right)\leq\mathbb{E}\left(\sup_{t,s}|X^{\prime}(t)-X^{\prime}(s)|\right).$

Then

$\displaystyle\mathbb{E}\left(\sup_{t,s}|Y(t)-Y(s)|\right)$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left(\sup_{t,s}|Y^{\prime}(t)-Y^{\prime}(s)|\right)\leq
2\mathbb{E}\left(\sup_{t,s}|X^{\prime}(t)-X^{\prime}(s)|\right)$
$\displaystyle\leq$ $\displaystyle
4\mathbb{E}\left(\sup_{t,s}|X(t)-X(s)|\right).$

Using the comparison principle, it follows that for $f,g\in C(\mathbb{T})$, if

$\mathbb{E}(|X_{f}(t)-X_{f}(s))|^{2}\leq\mathbb{E}(|X_{g}(t)-X_{g}(s)|^{2},$

then

$\mathbb{E}\left(\sup_{s,t}|X_{f}(t)-X_{f}(s))|\right)\leq
4\mathbb{E}\left(\sup_{s,t}|X_{g}(t)-X_{g}(s))|\right).$

This implies that if $g\in\mathscr{P}(\xi)$ then $f\in\mathscr{P}(\xi)$. It is
now easy to use the same argument as in the paper by Pisier [8] to obtain the
complete proof of Theorem 1.

In the case where the probability measure $\mu$ is absolutely continuous with
density $\varphi$, that is, $d\mu(u)=\varphi(u)du$, it is clear that if
$\varphi$ is bounded, then $\mathscr{P}\subset\mathscr{P}(\mu)$ where
$\mathscr{P}$ is the original Pisier algebra (associated to the Lebesgue
measure on $\mathbb{T}$).

###### Theorem 2

Let $\mu$ be a Borel probability measure on $\mathbb{T}$. If there is an
interval $I\subset\mathbb{T}$ and a constant $C>0$ such that for any
continuous function $g\geq 0$ on $\mathbb{T}$

$\int_{I}g(u)du\leq C\int_{I}g(u)d\mu(u),$

then $\mathscr{P}(\mu)\subset\mathscr{P}.$ In particular if $\mu$ admits a
density $\varphi$ that is continuous and strictly positive on some open
interval $I$ of $\mathbb{T}$, then $\mathscr{P}(\mu)\subset\mathscr{P}.$

  * Proof

Clearly there is a finite set of translations of $I$ that covers $\mathbb{T}$,
that is, there exist $t_{1},t_{2},\ldots,t_{N}$ in $\mathbb{T}$ such that
$\mathbb{T}\subset(I+t_{1})\cup\ldots\cup(I+t_{N}).$ Set
$\mu^{\prime}=(\sum_{k=1}^{N}\mu_{j})/N$ with $\mu_{j}=\delta_{t_{j}}*\mu$.
(Since each $\mu_{j}$ is a probability measure then $\mu^{\prime}$ is also a
probability measure.) Then for any continuous function $g\geq 0$ on
$\mathbb{T}$,

$\displaystyle\int_{\mathbb{T}}g(u)du$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{N}\int_{I+t_{j}}g(u)du=\sum_{j=1}^{N}\int_{I}g(u+t_{j})du\leq
C\sum_{j=1}^{N}\int_{I}g(u+t_{j})d\mu(u)$ $\displaystyle=$ $\displaystyle
C\sum_{j=1}^{N}\int_{I}g(u)d\mu_{j}(u)=CN\int_{I}g(u)d\mu^{\prime}(u)$
$\displaystyle\leq$ $\displaystyle CN\int_{\mathbb{T}}g(u)d\mu^{\prime}(u).$

In particular for any $t,s\in\mathbb{T}$, $f\in C(\mathbb{T})$,

$\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}du\leq
CN\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}d\mu^{\prime}(u).$

This implies by the comparison principle that
$\mathscr{P}(\mu^{\prime})\subset\mathscr{P}$. Moreover
$\mathscr{P}(\mu^{\prime})=\mathscr{P}(\mu)$. Indeed, for any
$t_{0}\in\mathbb{T}$, it is clear that
$\mathscr{P}(\mu)=\mathscr{P}(\delta_{t_{0}}*\mu)$ since the corresponding
processes $X_{f}$ are obtained from the other by a time translation by
$t_{0}$. This implies that $\mathscr{P}(\mu^{\prime})\subset\mathscr{P}(\mu)$.
To see that $\mathscr{P}(\mu)\subset\mathscr{P}(\mu^{\prime})$ also holds,
consider independent processes $X_{f}^{1},X_{f}^{2},\ldots,X_{f}^{N}$ such
that $X_{f}^{j}$ is the process associated to the measure $\delta_{t_{j}}*\mu$
for all $1\leq j\leq N.$ Set $X^{\prime}_{f}=(X_{f}^{1}+\ldots+X_{f}^{N})/N$.
Clearly $X^{\prime}_{f}$ can be viewed as the process associated to the
measure $\mu^{\prime}$ and

$\mathbb{E}(\sup_{s,t}|X^{\prime}_{f}(t)-X^{\prime}_{f}(s)|)\leq\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}(\sup_{s,t}|X^{j}_{f}(t)-X^{j}_{f}(s)|)=\mathbb{E}(\sup_{s,t}|X_{f}(t)-X_{f}(s)|)$

since each $X^{j}_{f}$ has the same distribution as $X_{f}$ (for all $j$). The
comparison principle yields
$\mathscr{P}(\mu)\subset\mathscr{P}(\mu^{\prime})$.

#### Remark 1:

If $\mu$ is the spectral measure of a stationary complex Gaussian process
$(\xi_{n})$ and $\mu$ has a density $\varphi$ such that $\varphi\in L^{p}(du)$
for some $1<p<\infty$, then for any $f\in\mathscr{P}(\mu)$ and
$t\in\mathbb{T}$, the formal Fourier random series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{int}$ converges in
$L^{2}(\mathbb{P})$. Indeed, for $t\in\mathbb{T}$ let $U:\mbox{span}(e^{2\pi
nt})\to\mbox{span}(\xi_{n})$ be the linear map such that $U(e^{2\pi
nit})=\xi_{n}$ for all $n\in\mathbb{Z}$. By relation (1) $U$ defines an
isometry from $L^{2}(\mu)$ to $L^{2}(\mathbb{P})$. Then by Hölder’s inequality
$L^{r}(du)\subset L^{2}(\mu)$ for some $1<r<\infty$ and hence $U$ is bounded
from $L^{r}(du)$ to $L^{2}(\mathbb{P})$. Since the Fourier transform of any
$f\in L^{r}(du)$ converges in $L^{r}(du)$ it follows that for any $f$ in
$L^{r}(du)$ the series $\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}$ converges in
$L^{2}(\mathbb{P})$. Applying this to the translates $f_{t}(u)=f(t+u)$ of $f$
yields that the random Fourier series
$\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi nit}$ converges in
$L^{2}(\mathbb{P})$ for every $t\in\mathbb{T}$.

##  3 Banach algebras spanned by the fractional Gaussian noise

###  3.1 Spectral function density of fractional Gaussian noise

Now we provide explicit examples of Pisier-type Banach algebras
$\mathscr{P}(\mu)$. The classical discrete complex fractional Gaussian noise
(fGn) of parameter $0\leq H<1$ is the complex Gaussian process
$\Delta=(\Delta_{n})_{n\in\mathbb{Z}}$ with mean $0$ and covariance function:

$\displaystyle\mathbb{E}(\Delta_{n}\overline{\Delta_{m}})=\gamma(n-m)={\scriptstyle\frac{1}{2}}|n-m+1|^{2H}+{\scriptstyle\frac{1}{2}}|n-m-1|^{2H}-|n-m|^{2H},\,\,\,m,n\in\mathbb{Z}.$
(2)

The process $\Delta$ can be realised as
$\Delta_{n}=(\delta^{1}_{n}+i\delta^{2}_{n})/\sqrt{2}$ where
$\delta^{1}=(\delta^{1}_{n})$ is the classical real fractional Gaussian noise
and $\delta^{2}$ is an independent copy of $\delta^{1}$. (Here
$\mathbb{E}(\delta^{1}_{n}\delta^{1}_{m})=\gamma(n-m)$.) The process $\Delta$
reduces to a sequence of i.i.d complex Gaussian random variables for $H=1/2$
but generally for $H\neq 1/2$ the random variables $(\Delta_{n})$ are
interdependent. (Fractional Gaussian noise is an important model with wide
range applications in physics and signal processing.) It is well-known that
the sequence $\Delta$ has a spectral density function and we shall denote it
by $\varphi_{H}$. The covariance function (2) implies that for any
$\ell\in\mathbb{Z}$, the sequence $(\Delta_{\ell+k})_{k\in\mathbb{Z}}$ has the
same distribution as the initial sequence
$\Delta=(\Delta_{k})_{k\in\mathbb{Z}}$, that is the distribution of $\Delta$
is stationary. Moreover for any $\ell\in\mathbb{Z}$, the sequence
$\Delta^{\ell}=(\Delta^{\ell}_{k})_{k\in\mathbb{Z}}$ defined by

$\displaystyle\Delta^{\ell}_{k}=\frac{1}{\ell^{H}}\left(\sum_{k\ell\leq
j<(k+1)\ell}\Delta_{j}\right),\,\,\mbox{ for all }k\in\mathbb{Z}$

has the same distribution as the initial sequence
$\Delta=(\Delta_{k})_{k\in\mathbb{Z}}$. This means that the distribution of
$\Delta$ is self-similar. (See Sinai [10] for some details on self-similar
distributions.) This can be seen by noting that the fGn $(\Delta_{k})$ is a
sequence of the increments $(X(k+1)-X(k))$ of the fractional Brownian motion
of index $H$ and it is well-known that it is a self-similar process in the
sense that, for any $\alpha>0$, the processes $\\{X(\alpha
t):t\in\mathbb{R}\\}$ has the same distribution as the process
$\\{\alpha^{H}X(t):t\in\mathbb{R}\\}$. Indeed,

$\Delta^{\ell}_{k}=\frac{1}{\ell^{H}}\left(X((k+1)\ell-X(k\ell)\right)\sim
X(k+1)-X(k)=\Delta_{k}$

by the self-similarity of fractional Brownian motion. Since the distribution
of $\Delta$ is stationary and self-similar, then by a result of Sinai [10,
Theorem 2.1], it admits a spectral density function $\varphi_{H}$ on the unit
circle $\mathbb{T}$ given by

$\displaystyle\varphi_{H}(t)=C(H)|e^{2\pi
it}-1|^{2}\left(\sum_{n=-\infty}^{\infty}\frac{1}{|t+n|^{2H+1}}\right),\,t\in\mathbb{T},t\neq
0$ (3)

where $C(H)$ is a normalising constant. Clearly,

$\displaystyle\varphi_{H}(t)$ $\displaystyle=$ $\displaystyle
4C(H)\left(\sin^{2}\pi
t\right)\sum_{n=0}^{\infty}\left(\frac{1}{(n+t)^{2H+1}}+\frac{1}{(n+1-t)^{2H+1}}\right)$
(4) $\displaystyle=$ $\displaystyle 4C(H)\left(\sin^{2}\pi
t\right)(\zeta(2H+1,t)+\zeta(2H+1,1-t))$

where $\zeta(.,.)$ is the classical Hurwitz zeta function. For $H=1/2$,
obviously $\varphi_{H}=1$ (constant function on $\mathbb{T}$). For $H\neq
1/2$, $\varphi_{H}$ is clearly continuous on $(0,1)$. Moreover it is clear
that $\zeta(2H+1,t)=O(t^{-(2H+1)})$ for $t$ near $0$ and
$\zeta(2H+1,1-t)=O(1-t)^{-(2H+1)}$ for $t$ near $1$. Since also $\sin^{2}\pi
t=O(t^{2}(1-t)^{2})$ for $t$ near $0$ or $1$, it follows from (4) that

$\displaystyle\varphi_{H}(t)=O(t^{1-2H}(1-t)^{1-2H}),\,\,\mbox{ for }t\mbox{
near }0\mbox{ or }1.$ (5)

This implies that for $H\leq 1/2$, the function $\varphi_{H}$ is continuous on
the whole unit circle $\mathbb{T}$ and for $H>1/2$, $t=0$ and $t=1$ are
singular points (actually $0=1$ in $\mathbb{T}$). However even in the case
where $H>1/2$, $\varphi_{H}$ is integrable and in fact (5) implies that

$\displaystyle\varphi_{H}\in L^{p}(\mathbb{T})\mbox{ for all
}p<1/(2H-1),\,\,H>1/2$ (6)

###  3.2 The Banach algebra $\mathscr{P}(H)$

The class $\mathscr{P}(\mu_{\Delta})$ where $\mu_{\Delta}$ is the spectral
measure of the discrete fractional Gaussian noise $\Delta=(\Delta_{n})$ of
parameter $H$ will simply be denoted $\mathscr{P}(H)$. Its study was initiated
in [5] where it was proven that for $0\leq H\leq 1/2$,
$\mathscr{P}(\frac{1}{2})\subset\mathscr{P}(H)$ and it was conjectured that
$\mathscr{P}(H)\subset\mathscr{P}(\frac{1}{2})$ for $\frac{1}{2}\leq H<1.$ The
question whether $\mathscr{P}(H)$ is in general a Banach algebra with the
usual multiplication of functions was also raised. With the discussion above,
it is now clear that for all $0\leq H<1$, $\mathscr{P}(H)$ is a homogeneous
Banach algebra on which all 1-Lipschitzian functions operate. Moreover for
$0\leq H<1/2$, the continuity and boundedness of $\varphi_{H}$ and the fact
that $\varphi_{H}(t)>0$ everywhere on $\mathbb{T}$ (except for $t=0=1$)
implies by Theorem 2 that
$\mathscr{P}(H)=\mathscr{P}(\frac{1}{2})=\mathscr{P}$. Finally for $H$ varying
in the interval $[0,1)$, $\mathscr{P}(H)$ is a decreasing family of
homogeneous Banach algebra. That is,

###### Theorem 3

For any $0\leq H_{1}\leq H_{2}<1$,
$\mathscr{P}(H_{2})\subset\mathscr{P}(H_{1}).$

  * Proof

This follows from the structure of the spectral density function $\varphi_{H}$
of the fractional Gaussian noise: For for $H_{1}\leq H_{2}$, it is the case
that the quotient $\varphi_{H_{1}}/\varphi_{H_{2}}$ is bounded. Indeed, write

$\frac{\varphi_{H_{1}}}{\varphi_{H_{2}}}=K\,\frac{(\zeta(1+2H_{1},t)+\zeta(1+2H_{1},1-t))}{(\zeta(1+2H_{2},t)+\zeta(1+2H_{2},1-t))},\,\,t\in(0,1)$

with $K=C(H_{1})/C(H_{2})$. Clearly $\varphi_{H_{1}}/\varphi_{H_{2}}$ is
continuous in the open interval $(0,1)$. For $t$ in a neighborhood of $0$ in
$\mathbb{T}$,

$\frac{\varphi_{H_{1}}}{\varphi_{H_{2}}}=O(t^{-(1+2H_{1})}/t^{-(1+2H_{2})})=O(t^{2(H_{2}-H_{1})}),$

which implies that $\lim_{t\to 0}\varphi_{H_{1}}(t)/\varphi_{H_{2}}(t)=0.$
Similarly for $t$ near $1$, $\lim_{t\to
1}\varphi_{H_{1}}(t)/\varphi_{H_{2}}(t)=0.$ This implies that
$\varphi_{H_{1}}/\varphi_{H_{2}}$ is continuous in the closed interval $[0,1]$
and hence $\varphi_{H_{1}}/\varphi_{H_{2}}$ is continuous and bounded on the
unit circle $\mathbb{T}$. Then for a constant $M$ depending only on $H_{1}$
and $H_{2}$ we have that

$\displaystyle\varphi_{H_{1}}(t)\leq M\,\varphi_{H_{2}}(t)\mbox{ for all
}t\in\mathbb{T}.$ (7)

Then for each $f\in C(\mathbb{T})$,

$\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}\varphi_{H_{1}}(u)du\leq
M\int_{\mathbb{T}}|f(t+u)-f(s+u)|^{2}\varphi_{H_{2}}(u)du$

which yields by the comparison principle that
$\mathscr{P}(H_{2})\subset\mathscr{P}(H_{1})$.

##  4 Boundedness of random Fourier series

###### Theorem 4

Let $\xi=(\xi_{n})_{n\in\mathbb{Z}}$ be a stationary complex Gaussian process
with mean $0$, unit variance, covariance matrix
$(\gamma(n-m))_{n,m\in\mathbb{Z}}$ and such that
$\mathbb{E}(\xi_{n}\xi_{m})=0$ for all $n,m\in\mathbb{Z}$. Assume that there
exists a constant $b\geq 0$ such that the operator defined in
$\ell^{2}(\mathbb{Z})$ by the bi-infinite matrix $(a_{nm})_{n,m\in\mathbb{Z}}$
given by

$a_{nm}=|\gamma(n-m)||nm|^{-b}\mbox{ for }n\neq 0,\,m\neq 0\mbox{ and
}a_{nm}=0\mbox{ for }n=m=0$

is bounded. Let $f\in L^{2}(\mathbb{T})$. If

$\displaystyle\sum_{|n|\geq
2}\frac{\left(\sum_{|k|\geq|n|}|\hat{f}(k)|^{2}|k|^{2b}\right)^{1/2}}{|n|(\log|n|)^{1/2}}<\infty,$
(8)

then the random series $t\mapsto\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi
nit}$ is bounded almost surely.

In particular if $f\in C(\mathbb{T})$ and satisfies (8) then
$f\in\mathscr{P}(\mu)$ where $\mu$ is the spectral measure of $\xi$.

  * Proof

Condition (8) is a generalisation of the following condition for independent
variables: For a sequence $(g_{n})$ of independent standard Gaussian random
variables and for $f\in C(\mathbb{T})$ and a sequence $(a_{n})_{n\geq 1}$ of
real numbers, if

$\sum_{n=2}^{\infty}\frac{\left(\sum_{k=n}^{\infty}a_{k}^{2}\right)^{1/2}}{n(\log
n)^{1/2}}<\infty,$

then the random series $t\mapsto\sum_{n=1}^{\infty}a_{n}g_{n}e^{2\pi nit}$,
$t\in\mathbb{T}$, converges uniformly. (See Marcus and Pisier [4, p. 122].)
Consider the series
$S(f,\xi,t)=\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}\exp(2\pi int)$,
$t\in\mathbb{T}$ and write for $N\in\mathbb{N}$

$S_{N}(t)=\sum_{|n|\leq N}\hat{f}(n)\xi_{n}e_{n}(t),\,\,\mbox{ with
}e_{n}(t)=\exp(2\pi int),\,\,\mbox{ for all }n$

and $\gamma(n-m)=\mathbb{E}(\xi_{n}\overline{\xi_{m}})$. Then for $N\leq M$ in
$\mathbb{N}$,

$\displaystyle\mathbb{E}(|S_{M}(t)-S_{N}(t)|^{2})$ $\displaystyle\leq$
$\displaystyle\sum_{N<|n|\leq
M}|\hat{f}(n)|^{2}|e_{n}(t)|^{2}+\sum_{N<|n|\neq|m|\leq
M}|\gamma(n-m)||\hat{f}(n)||\hat{f}(m)||e_{n}(t)||e_{m}(t)|$
$\displaystyle\leq$ $\displaystyle\sum_{N<|n|\leq
M}|\hat{f}(n)|^{2}|e_{n}(t)|^{2}+\sum_{N<|n|\neq|m|\leq
M}a_{nm}|n|^{b}|m|^{b}|\hat{f}(n)||\hat{f}(m)||e_{n}(t)||e_{m}(t)|.$

Now since the matrix $A=(a_{nm})$ defines a bounded operator in
$\ell^{2}(\mathbb{Z})$, then

$\sum_{N<|n|\neq|m|\leq
M}a_{nm}|nm|^{b}|\hat{f}(n)||\overline{\hat{f}(m)}|e_{n}(t)||e_{m}(t)|\leq
K\sum_{N<n\leq M}|\hat{f}(n)|^{2}|n|^{2b}|e_{n}(t))|^{2}$

for some constant $K>0$. (In fact we may write this sum as $wA\overline{w}$
where $w=(w_{n})_{n\in\mathbb{Z}}$ and $w_{n}=|\hat{f}(n)||n|^{b}|e_{n}(t)|$
for $N<n\leq M$ and $w_{n}=0$ outside the interval $(N,M]$ in $\mathbb{Z}$ and
since $A$ is positive semidefinite and bounded, then $wA\overline{w}\leq
K\|w\|^{2}$.) It follows that

$\mathbb{E}(|S_{M}(t)-S_{N}(t)|^{2})\leq K^{\prime}\sum_{N<n\leq
M}|\hat{f}(n)|^{2}|n|^{2b}|e_{n}(t))|^{2}$

for some constant $K^{\prime}>0$. Now clearly inequality (8) implies in
particular that for each $t\in\mathbb{T}$ the series
$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}|n|^{2b}|e_{n}(t)|^{2}$ converges from
which it follows that
$\lim_{N,M\to\infty}\mathbb{E}(|S_{M}(t)-S_{N}(t)|^{2})=0$. This yields the
$L^{2}$–convergence of the series $S(f,\xi,t)$. The same argument yields that
for all $t,s\in\mathbb{T}$,

$\mathbb{E}|S(f,\xi,t)-S(f,\xi,s)|^{2}\leq
K\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{2}|n|^{2b}|e_{n}(t)-e_{n}(s)|^{2}$

for some constant $K>0$. That is

$\mathbb{E}|S(f,\xi,t)-S(f,\xi,s)|^{2}\leq
K\mathbb{E}\left|\sum_{n\in\mathbb{Z}}\hat{f}(n)|n|^{b}g_{n}(e_{n}(t)-e_{n}(s))\right|^{2}$

where $(g_{n})$ is the standard complex Gaussian sequence. Inequality (8)
implies that the random series
$t\mapsto\sum_{n\in\mathbb{Z}}\hat{f}(n)|n|^{b}g_{n}e_{n}(t)$ converges
uniformly on $\mathbb{T}$. This yields by the comparison principle that the
paths $t\to S(f,\xi,t)$ are bounded almost surely. If in addition $f$ is
continuous then $f\in\mathscr{P}(\xi)$.

An example of a covariance function $\gamma$ for which the matrix
$(\gamma(n-m)|nm|^{-b})$ defined a bounded operator in $\ell^{2}(\mathbb{Z})$
is obtained by assuming $|\gamma(n-m)|\leq K|n-m|^{-a}$ for some constants
$K>0$ and $0<a<1.$ This can be obtained by using the classical Schur test
together with the following well-known fact: for real numbers $0<a<1$ and
$d>0$ such that $a+d>1$, there exists a constant $K>0$ (depending on $a$ and
$d$) such that

$\mbox{ for all }n\in\mathbb{Z},\sum_{m\neq n}|n-m|^{-a}|m|^{-d}\leq
K|n|^{1-(a+d)}.$

Set $a_{nm}=|n-m|^{-a}|nm|^{-b}$ (with $a_{nn}=a_{n,0}=a_{0,n}=0$) and
consider a real number $c>0$ such that $a+b+c>1$ and
$x=(x_{n})_{n\in\mathbb{Z}}$ with $x_{n}=|n|^{-c}$ for $\neq 0$ and $x_{0}=1$.
It is the case that

$\forall n\,\,\,\sum_{m\in\mathbb{Z}}a_{nm}x_{m}\leq Kx_{n}\,\,\,\mbox{ and
}\,\,\,\forall m\,\,\,\sum_{n\in\mathbb{Z}}x_{n}a_{nm}\leq Kx_{m}.$

Hence Schur’s test yields that the operator defined by $(a_{nm})$ is bounded.
In particular the covariance function $\gamma_{H}$ of the fractional Gaussian
noise of index $H$ is such that the corresponding matrix $(a_{nm})$ given by
$a_{nm}=\gamma_{H}(n-m)|nm|^{-b}$ for any $b>0$ defines a bounded operator.
Indeed it is easy to verify that $0<\gamma_{H}(n-m)\leq|n-m|^{-2(1-H)}$ for
$m\neq n.$

###### Corollary 1

Let $f\in L^{2}(\mathbb{T})$ and $\frac{1}{2}\leq H<1$. If there exist
$\alpha>H$ and $K>0$ such that

$|\hat{f}(n)|\leq K|n|^{-\alpha},\,\,\mbox{ for all }n\neq 0,$

then the function $t\mapsto\sum_{n\in\mathbb{Z}}\hat{f}(n)\Delta_{n}e^{2\pi
int}$ is bounded almost surely. In particular if in addition $f\in
C(\mathbb{T})$ but the series $\sum_{n\in\mathbb{Z}}|\hat{f}(n)|$ diverges,
then $f\in\mathscr{P}(H)$ but $f\notin A(\mathbb{T}).$

This follows from the fact that by taking $b=H-1/2>0$ it is clear that
inequality (8) holds.

#### Remark 2:

It would be interesting to investigate whether inequality (8) implies the
uniform convergence (almost surely) of the random Fourier series
$t\to\sum_{n\in\mathbb{Z}}\hat{f}(n)\xi_{n}e^{2\pi nit}$ as it is the case for
i.i.d random variables. Pisier [9] obtained that the Banach space of $f\in
L^{2}(\mathbb{T})$ such that $\sum_{n\in\mathbb{Z}}Z_{n}\hat{f}(n)e^{2\pi
int}$ (with $(Z_{n})$ i.i.d Gaussian variables) is bounded a.s. is of cotype
2. It would be interesting to consider this question for the general
stationary Gaussian process $(\xi_{n})$. Pedersen [7] proved that spectral
synthesis holds in $\mathscr{P}$. This question is also of some interest in
the general setting discussed in this paper.

#### Acknowledgments:

I would like to thank the anonymous referee for very helpful comments,
suggestions and new results that have significantly improved this paper.
Theorem 2 was directly suggested by the referee. This research received with
gratitude funding from the College of Economics and Management Sciences of the
University of South Africa and the EU project CID. Most of this work was
conducted during a visit to the National Institute for Research in Digital
Science and Technology (INRIA) in Nancy. I thank Mathieu Hoyrup for the
invitation and his hospitality.

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