Sevasti Karypi
email: sevikkar1@gmail.com

Educational Robotics Application in Primary and Secondary Education: A Challenge for the Greek Teachers Society

Keywords: Educational Robotics, schools, teachers, ER integration, Greek schools. JEL Classification: Ι21, I29

This paper presents the results of a postgraduate study that was designed to investigate the attitudes and views of Greek teachers in primary and secondary education on the application of Educational Robotics (ER), towards the goal of drawing useful insights on how it can be further integrated in Greek schools. A total of 70 teachers participated in this study, currently working for primary and secondary schools in Greece and being involved in ER projects and ER-related activities. According to the research findings, ER has significant benefits for students and educators, as it fosters positive attitudes towards STEM education, encourages independent and active learning, facilitates teaching, and provides opportunities for the development of cognitive, social and communication skills. However, factors such as the lack of funding and physical infrastructure, the inadequate training of teachers and curriculum scheduling inflexibility, hinder its application. Thus, several structural and procedural actions should be taken in order to further integrate ER in Greek schools.

INTRODUCTION

Educational Robotics (ER) was developed over the past few decades as an innovative tool to promote STEM and aims to better prepare students for the knowledge and information society of the future. Initially, ER was developed according to the theory of constructivism [START_REF] Mikropoulos | Educational robotics as mind tools[END_REF], according to which the mind responds to stimuli of the external environment and learning is a continuous process based on the student's synthetic ability and facilitated by the teacher [START_REF] Devries | Developing Constructivist Early Childhood Curriculum: Practical Principles and Activities[END_REF]. Later on, Papert with his work (1980) influenced its evolution both by developing the Logo programming language and by advocating the theory of constructionism. According to this theory, knowledge acquisition is more effective when teachers engage in building of artifacts that the students can relate with. Other theories linked with ER are active learning, learning through making (maker movement) [START_REF] Goldman | Using educational robotics to engage inner-city students with technology[END_REF], and educational entertainment (edutainment) [START_REF] Atmatzidou | LEGO Mindstorms application in Elementary and High School.Game as a learning trigger[END_REF]. Recent research has demonstrated that the application of ER in various educational settings has many benefits for students in terms of cognitive, social and communication abilities' development, as well in forming positive attitudes towards STEM. As regards cognitive abilities, it has been found thatER fosters students' creative and critical thinking, independent and active learning, and decision making [START_REF] Barak | Robotics projects and learning concepts in science, technology and problem solving[END_REF][START_REF] Khanlari | Effects of robotics on 21st century skills[END_REF], encourages cognitive problem solving, mental processing and logical sequencing [START_REF] Lindh | Does Lego training stimulate pupils' ability to solve logical problems[END_REF][START_REF] Kazakoff | The effect of a classroom-based intensive robotics and programming workshop on sequencing ability in early childhood[END_REF], and supports intrinsic motivation and project management skills [START_REF] Highfield | Robotic toys as a catalyst for mathematical problem solving[END_REF][START_REF] Hong | Collaborative learning in technological project design[END_REF]. It also supports students in learning of abstract concepts [START_REF] Whittier | Teaching evolution to non-English proficient students by using LEGO robotics[END_REF], and second language acquisition (Chen et al, 2011). In addition, it has been documented that ER positively affects students attitudes towards STEM and school performance in related subjects [START_REF] Barker | Robotics as means to increase achievement scores in an informal learning environment[END_REF][START_REF] Bers | Computational thinking and tinkering: Exploration of an early childhood robotics curriculum[END_REF], and other communication and social skills by encouraging teamwork and two-way communication (Ruiz & Aviles, 2004). A limited number of researchers have also investigated teachers' attitudes towards ER. In particular, it has been found that teachers think that integrating ER in the educational process can bring significant benefits for students regarding their cognitive and social skills [START_REF] Khanlari | Teachers' perceptions of the benefits and the challenges of integrating educational robots into primary/elementary curricula[END_REF], as well for supporting in-class teaching [START_REF] Fridin | Acceptance of socially assistive humanoid robot by preschool and elementary school teachers[END_REF], particularly in STEM [START_REF] Reich-Stiebert | Robots in the classroom: What teachers think about teaching and learning with education robots[END_REF]. In Greece, [START_REF] Theodoropoulos | Teacher and student views on educational robotics: The Pan-Hellenic competition case[END_REF] found that teachers involved in ER projects consider that the latter are very beneficial for students, who are able to develop their creativity, innovative thinking, communication and project management skills, problem solving, self-confidence and self-discipline. In the same research it was found that the most important barriers for integrating ER in schools are the lack of time and insufficient financial resources available to schools for implementing such projects [START_REF] Theodoropoulos | Teacher and student views on educational robotics: The Pan-Hellenic competition case[END_REF]. In many countries, ER has been a popular and well-organized extra-curricular activity often taking place after the end of the regular academic day [START_REF] Sullivan | Robotics and discovery learning: pedagogical beliefs, teacher practice, and technology integration[END_REF]. As far as the subject concerned, ER is integrated in computer and technology education, as well as in non-technical education, such as languages. In the first case, the primary goal is to provide knowledge about robots and technology, by introducing students in learning objects related to computer science and programming. According to [START_REF] Balch | Designing personal robots for education: Hardware, software, and curriculum[END_REF], such a lesson typically includes an initial introduction to robot programming and, subsequently, the practical training of this knowledge in robotics' construction, which results in building a sense of ownership and strengthening interest of the student [START_REF] Mubin | Improving speech recognition with the robot interaction language[END_REF]). In the case on non-technical education, ER is used as means of teaching various learning objects, such as mathematics and geometry [START_REF] Highfield | Early mathematics learning through exploration with programmable toys[END_REF], as well as second languages [START_REF] Kanda | Interactive robots as social partners and peer tutors for children: A field trial[END_REF]. In this case, the most frequent use of ER is STEM-related subjects, aimed at interdisciplinary science learning, enhancing student literacy and increasing the number of students wishing to pursue a professional or academic course in the relevant disciplines [START_REF] Mataric | Materials for Enabling Hands-On Robotics and STEM Education[END_REF]. However, it should be noted that ER does not apply exclusively to STEM fields but also to other subjects such as literature, history, social sciences, dance, music and arts, providing students the opportunity to find new ways of cooperation, expression, innovation and critical thinking. In history, for example, with the construction of a catapult robot, children had the opportunity to experience the development of the technology of that era and the work of Archimedes. An interdisciplinary approach, therefore, is best suited to exploiting the benefits of integrating ER in the educational process [START_REF] Eguchi | Educational robotics for promoting 21st century skills[END_REF]. Furthermore, the development of many ER commercial packages, with particularly improved and user-friendly design (eg. LEGO Mindstorms), have given the opportunity to organize specific projects in schools that have proven to be extremely effective in terms of enriching the learning process, motivating students and developing new skills (Detsikias & Alimisis, 2011). It should also be noted that ER integration in education has also been promoted by several robotics competitions organized at international or national level and aimed mainly at high school students, while offering significant opportunities for interaction, cognitive and social development, and improved learning in STEM-related fields [START_REF] Nugent | The Impact of Educational Robotics on Student STEM Learning, Attitudes, and Workplace Skills[END_REF]. At the same time, over the last few years, various actions, activities and events are taking place in Europe concerning ER development addressed to teachers, students and other specialists, including thematic workshops (eg. International Workshop "Teaching Robotics, Teaching with Robotics"), conferences (eg."Robotics in Education"), training seminars for teachers (eg.TERECoP, Roberta Teacher Training) and other local or regional networks (eg. RobotScuola in Italy and CENTREBOT in Austria). However, in Greece, ER has been limited and often a result of the initiative of certain teachers and individuals. Indeed, although the application of ER in the lower educational grades of mainstream schools is primarily occasional, given the lack of information and training programs for teachers on this subject, there are several remarkable cases of autonomous projects with the initiative of individual teachers, which have successfully participated in international and domestic competitions. An extremely important role in the development of the field of ER in Greece is played by the nonprofit organization WRO Hellas, which organizes national ER competitions and aims to the development of STEM education in Greek schools. Given that ER in Greece has been around for less than 10 years, little is known about its how it is used, its benefits, and how it is perceived by the educators in terms of ER integration in the educational process, including respective barriers. The purpose of this study is to fill this gap by presenting and analyzing data that were collected by surveying a random sample of 70 teachers from primary and secondary schools all over Greece. Though all these teachers had used ER in their teaching, the degree to which they did varied from teacher to teacher. The research was conducted using Google forms between 4-12 December 2017.

RESEARCH QUESTIONS

The aim of this study is to understand the current state of ER in Greece, by assessing the experiences of the teachers who participate in this development and evaluating their views and attitudes towards ER. In particular, this study aims to investigate teachers' views regarding: (a) the current state of ICT integration and implementation in Greek mainstream schools, (b) the benefits for students involved in ER projects, (c) the educational subjects that teachers consider more appropriate for the implementation of ER, (d)the most significant limitations and barriers observed while planning and organizing ER projects in schools, and (e) ER further implementation and potential in primary and secondary education in Greece.

RESEARCH SAMPLE

In this study, a convenience sampling method was used, so as to collect a representative set of teachers, given also the limited number of educators involved in ER projects in Greece. The sample of the survey consists of 70 primary and secondary education teachers from all over Greece, and collections of data was performed through the Google forms platform with a 64.5% response rate, which is considered to be satisfactory. The teachers involved in the research have applied ER in class and / or have taken part in robotics competitions. Teachers' e-mail addresses were searched through the school network, through social networking tools, and through personal contacts of the researcher during ER competitions, in which Greek primary and secondary schools took part or/and awarded. As regards the demographic profile of participants, it has been observed that the sample was evenly distributed concerning their gender. The 35,7% of participants were graduates of Technical Institutions or Universities, 57,1% were postgraduates and 7% were PhD holders. Also, the 65,7% were permanently appointed, 10,0% were temporarily transferred, 2,9% were replenishing, while 14,3% were in higher hierarchical positions (school principals / vice principals). Regarding their specialization, the 60% of the sample is specialized in Information Technology, 4,3% in Physics, 2,9% in Chemistry, 5,7% in Engineering, and 27,1% in various other fields (mainly primary education teachers). The percentage of those having been trained in ICTs was exceptionally high, coming up to 91,4%. Finally, the average sample age was 45+/-6,9 years and the average service time was 19,7+/-6,1 years.

DATA COLLECTION AND ANALYSIS

Data collection was performed via a questionnaire that was developed by the researcher according to the findings of the relevant research literature concerning ER. The questionnaire consists of 5 parts and close-ended questions. The first part of the questionnaire reflects the demographic and professional characteristics of the respondents such as gender, age, years of service, their level of education, their position and skills, and whether they have ICT training. The second part of the research tool consists of five Likert-scale questions regarding ICT integration in schools that respondents are currently working for, and the third part involves ten questions concerning ER benefits. In the fourth part of the questionnaire, the level of robotics implementation in schools is first determined through 9 Likert scale questions, and then multiple choice question marks the courses that are considered more appropriate by teachers for the use of ER. Finally, in the fifth part of the research tool assesses the experience of respondents in ER, as well as their views regarding its further potential. A pilot study was also performed in a sample of 5 teachers so as to identify potential problems and assess its reliability. Statistical processing was performed with the statistical package SPSS 20.0.

RESULTS

As regards ICT integration in the educational process, it turns out to be relatively high (A.S. =3,60, S.D.=82). It should be noted that teachers participating in this study highly incorporate ICTs in their teaching, and they extensively use digital technologies both in the classroom as well as in other extra-curriculum activities. This finding is quite expected, as teachers involved in this survey are also involved in ER projects and have been trained in ICT-related subjects. On the other hand, the technological infrastructure of schools for implementing ICTs is considered mostly moderate, providing evidence that technical and technological resources of Greek schools are not adequate for developing ER projects and further incorporating robotics in the educational process. Use and integration of ICTs 3,60 0,82 Regarding the benefits of teaching robotics, this study revealed that ER has significant advantages both for students and educators, as it facilitates teaching in class and especially STEM-related subjects. More precisely, it was found that ER has a positive impact on improving students' mental and cognitive skills, while it also fosters self-expression, research interest, creativity, project management skills, communication and collaboration, problem-solving and selfconfidence. Findings in international studies confirm that the applications of robotics in various educational settings contribute to the development of mental skills and the acquisition of science-related knowledge [START_REF] Lindh | Does Lego training stimulate pupils' ability to solve logical problems[END_REF][START_REF] Barak | Robotics projects and learning concepts in science, technology and problem solving[END_REF][START_REF] Highfield | Robotic toys as a catalyst for mathematical problem solving[END_REF][START_REF] Hong | Collaborative learning in technological project design[END_REF][START_REF] Kazakoff | The effect of a classroom-based intensive robotics and programming workshop on sequencing ability in early childhood[END_REF][START_REF] Khanlari | Effects of robotics on 21st century skills[END_REF], while providing upgraded opportunities for team work, communication and collaboration (Ruiz & Aviles, 2004), and enhancing positive attitudes towards STEM [START_REF] Barker | Robotics as means to increase achievement scores in an informal learning environment[END_REF][START_REF] Bers | Computational thinking and tinkering: Exploration of an early childhood robotics curriculum[END_REF]. It was also found that teachers have positive attitudes towards ER integration in the educational process, as it has been suggested in other related studies internationally [START_REF] Fridin | Acceptance of socially assistive humanoid robot by preschool and elementary school teachers[END_REF][START_REF] Khanlari | Teachers' perceptions of the benefits and the challenges of integrating educational robots into primary/elementary curricula[END_REF]Reich-Stiebert&Eyssel, 2016), and in Greece [START_REF] Theodoropoulos | Teacher and student views on educational robotics: The Pan-Hellenic competition case[END_REF]. Of course, several barriers exist, especially as regards schools technological infrastructure and financial resources available for ER projects [START_REF] Theodoropoulos | Teacher and student views on educational robotics: The Pan-Hellenic competition case[END_REF], as also shown in this research. The findings of this research are of significant importance regarding ER integration in Greek schools. In particular, students' positive attitudes towards ER projects and applications imply that ER is of great potential for fostering STEM education in Greece. The systematic involvement of students in ER applications contributes to the enhancement of their interest in science and the increases of levels of intrinsic motivation for STEM. In addition, the teachers who participated in this study confirmed that robotics facilitates teaching, although it is worth noting that the views on whether the relevant applications lead to more positive attitudes towards theoretical subjects (eg. history) are mixed. This finding, of course, is expected if one considers the inherent link between robotics and STEM and the fact that the majority of applications are implemented in the corresponding lessons. Despite the above mentioned benefits, this study revealed that ER integration in Greek schools has many barriers, especially as regards the lack of sufficient time for designing such projects, the limited availability of technological resources, inadequate access to appropriate software / hardware, and the limited flexibility of the curriculum. At the same time, moderate seems to be the support of school administration for the integration of ER while the issue of insufficient digital literacy of teachers for the use of robotics is also significant. Indeed, schools' responsiveness to ER integration is not high, as documented in this research, implying that ER is a challenge for the Modern Greek teaching society. In this respect, there is a need to take action, both centrally and at school level, in order to remove obstacles and make it possible to integrate ER more effectively into the school reality. A particularly important variable that can work in this direction is the organization of training seminars at a central level so that teachers are able to use relevant applications. At a school level, schools and their administrations should take advantage of the training opportunities offered by the private sector, which has been actively involved in this field over the last period, with the example of WRO Hellas. In this context, partnerships between schools and private bodies should be welcomed by school administrations. Besides, a specific plan for informing teachers themselves could also bear fruit for them to be informed about the possibilities offered by ER and to take autonomous action. Finally, at central level, a dialogue should be opened on the possible integration of robotics into the curriculum, notably through cross-thematic approaches, by adopting the best practices currently being developed in other countries. This study has a number of limitations that need to be mentioned. The most important limitation is the relatively small sample of teachers participating in this research, although teachers involved in relevant projects and applications in Greece is quite limited. A further limitation concerns the fact that no attempt was made to investigate the specific characteristics of the ER projects in which the teachers are involved, so as to examine any links between these projects and teachers' respective views on benefits and possibilities for further implementation. Given this research limitations and the fact that ER is a growing and promising field in Greece, future research should further explore teachers' attitudes towards ER, including educators of non-STEM subjects who are not currently involved in ER projects. Furthermore, future research is needed as regards students' views on ER, in order to better understand ER benefits and limitations. Lastly, a matter of future research interest is the examination of the causal link between systematic child engagement with ER and STEM school performance, as well as the investigation of ER involvement impact on the academic and professional path of students in their latter adult life.

Table 1 :

 1 ICTs use and integration in schools

		None	A little	Moderate	A lot	Very Much
		Ν % Ν	% Ν	%	Ν	%	Ν	%	A.S. S.D.
	How adequate is
	technological
	infrastructure/ technological means at school curriculum (PC,	3 4,3% 9	12,9 %	2 3	32,9% 20	28,6 %	15	21,4 %	3,50 1,10
	software, technical
	support)?
	Up to what extent do you integrate ICTs in teaching process?	2 2,9% 4 5,7% 8 11,4% 25	35,7 %	31	44,3 %	4,13 1,02
	To what degree have you
	been using digital technologies in	4 5,7% 3 4,3% 7 10,0% 29	41,4 %	27	38,6 %	4,03 1,09
	classroom?
	How often do you use
	ICTs for other curricular activities outside the	1 1,4% 5 7,1%	1 0	14,3% 32	45,7 %	22	31,4 %	3,99 0,94
	classroom?
	To what extent do you use robotics in the process of teaching?	26	37,1 %	16	22,9 %	1 1	15,7% 12	17,1 %	5 7,1% 2,34 1,33

In addition, it is found that, according to teachers' views, the most appropriate subjects of the school curriculum where ER can be used are technology (80,0%), physics (62,9%), engineering (57,1%), mathematics, and geometry (52,9%). The application of robotics is also suggested to be used in independent projects as extra-curriculum activities (68,6%). On the other hand, history is a subject of limited potential for ER application. Factor analysis performed regarding teachers' satisfaction from their experience while participating in ER projects revealed two factors, i.e. their satisfaction with students' involvement and their satisfaction with schools response. In particular, as demonstrated by the average rating of these factors, students' response to the project development procedure in terms of participation, co-operation, expression of ideas and development of skills and positive attitudes is remarkably high (A.S.=4,20, S.D.=0,56). On the contrary, the response of school in terms of logistics infrastructure sufficiency and teaching staff support is low (A.S. =3,08, S.D. =0,94). Accordingly, it can be argued that teachers participating in ER projects in Greece consider students as highly involved in the related activities, while schools are not ready yet for implementing such projects, a finding that should be well considered by educators, school administrators and educational policy-makers. A remarkable finding concern also the moderate support provided by other teachers not involved in ER activities. Thus, the interdisciplinary approach suggested for ER integration in schools is not facilitated in the Greek educational setting due to both schools inadequate infrastructure and other organizational and human-related factors. The students co-operated harmoniously. 0 0,0% 1 2,4% 4 9,8% 23 56,1% 13 31,7% 4,17 0,70

The students were able to express their ideas and be creative. 0 0,0% 0 0,0% 3 6,8% 27 61,4% 14 31,8% 4,25 0,58

The students developed remarkable mental and communicational skills through the project. 0 0,0% 0 0,0% 7 15,2% 28 60,9% 11 23,9% 4,09 0,63

The students developed positive attitudes towards learning objects. 0 0,0% 0 0,0% Finally, the factor analysis revealed two factors regarding ER potential for further integration in schools and its application in Greek educational settings, i.e. structural and procedural actions. In particular, the need of structural action for further integration of robotics is schools is particularly stressed by teachers participating in this study (A.S.=4,37, S.D.=0,57), and secondly, the need of procedural actions is also pointed out (A.S.=3,73, S.D.=0,74). In addition, further integration of ER in Greek schools is also considered as necessary (A.S.=4,05, S.D.=0,54).

DISCUSSION

ER is a remarkably innovative teaching and educational application that provides the means for modernizing teaching and learning, having great benefits in terms of ICT integration in schools and curriculums. However, it is well known that up to now, there has not been a full alignment between the technological progress that has been achieved in the global society of knowledge and the respective school reality. Thus, the implementation of ER, though very promising, is still at a very primary level. The findings of this research suggest that, while ICTs are rapidly integrated in the Greek school setting, robotics still remain a relatively unexploited field, as it is also observed at an international level [START_REF] Mataric | Materials for Enabling Hands-On Robotics and STEM Education[END_REF][START_REF] Alimisis | Educational robotics: New challenges and trends[END_REF]. However, it should be also noted that the finding that ICTs seem to be well incorporated in Greek schools, as documented in this research, can be attributed to the fact that teachers participated in this study have been activated in the field of teaching robotics and therefore are familiar with modern technologies.


Jean Bertoin
email: jean.bertoin@math.uzh.ch

Scaling exponents of step-reinforced random walks

Keywords: Reinforcement, random walk, scaling exponent, heavy tail distribution. Mathematics Subject Classification: 60G50, 60G51, 60K35

Scaling exponents of step-reinforced random walks

Introduction

In 1955, Herbert A. Simon [START_REF] Simon | On a class of skew distribution functions[END_REF] introduced a simple reinforcement algorithm that runs as follows. Consider a sequence ε 1 , ε 2 , ε 3 , . . . in {0, 1} with ε 1 = 1. The n-th step of the algorithm corresponds to an innovation if ε n = 1, and to a repetition if ε n = 0. Specifically, denote the number of innovations after n steps by

σ(n) = n i=1 ε i for n ≥ 1,
and let also X 1 , X 2 , . . . denote a sequence of different items (in [START_REF] Simon | On a class of skew distribution functions[END_REF], these items are words). One constructs recursively a random sequence of items X1 , X2 , . . . by deciding that Xn = X σ(n) if ε n = 1, and that Xn = XU(n) if ε n = 0, where U (n) is random with the uniform distribution on [n -1] = {1, . . . , n -1} and U (2), U (3), . . . are independent.

Simon was especially interested in regimes where either ε n converges in Césaro's mean to some limit q ∈ (0, 1), which we will refer to as steady innovation with rate q, or σ(n) grows like1 n ρ for some ρ ∈ (0, 1) as n → ∞, which we will call slow innovation with exponent ρ. By analyzing the frequences of words with some fixed number of occurrences, he pointed out that these regimes yield a remarkable one-parameter family of power tail distributions, that are known nowadays as the Yule-Simon laws and arise in a variety of empirical data. This is also closely related to preferential attachment dynamics, see e.g. [START_REF] Bornholdt | World Wide Web scaling exponent from Simon's 1955 model[END_REF] for an application to the World Wide Web. Clearly, repetitions in Simon's algorithm should be viewed as a linear reinforcement, the probability that an item is repeated being proportional to the number of its previous occurrences.

In the present work, the items X 1 , X 2 , . . . are i.i.d. copies of some real random variable X, which we further assume to be independent of the uniform variables U (2), U (3), . . .. Picking up on a key question in the general area of reinforced processes (see notably the survey [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF] by Pemantle, and also some more recent works [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF][START_REF] Cotar | Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs[END_REF][START_REF] Disertori | Transience of edgereinforced random walk[END_REF][START_REF] Kious | Phase transition for the once-reinforced random walk on Z d -like trees[END_REF][START_REF] Sabot | Edge-reinforced random walk, vertexreinforced jump process and the supersymmetric hyperbolic sigma model[END_REF] and references therein), our purpose is to analyze how reinforcement affects the growth of partial sums. Specifically, we write

S(n) = X 1 + • • • + X n
for the usual random walk with step distribution X, and

Ŝ(n) = X1 + • • • + Xn
for its reinforced version, and we would like to compare Ŝ(n) and S(n) when n 1. The main situation of interest is when S has a scaling exponent α ∈ (0, 2], in the sense that

lim n→∞ n -1/α S(n) = Y in law, (1)
where Y denotes an α-stable variable. Recall that this holds if and only if the typical step X belongs to the domain of normal attraction (without centering) of a stable distribution, in the terminology of Gnedenko and Kolmogorov [START_REF] Gnedenko | Limit distributions for sums of independent random variables. Translated from the Russian, annotated[END_REF].

We shall refer to [START_REF] Aaronson | Characteristic functions of random variables attracted to 1-stable laws[END_REF] as an instance of α-diffusive asymptotic behavior, the usual diffusive situation corresponding to α = 2.

The asymptotic behavior of the step-reinforced random walk Ŝ has been considered previously in the literature when ε 2 , ε 3 , . . . are given by i.i.d. samples of the Bernoulli law with parameter q ∈ (0, 1). This is of course a most important case of a steady regime with innovation rate q a.s. It has been shown recently in [START_REF] Bertoin | Universality of noise reinforced Brownian motions[END_REF] that when X ∈ L 2 (P), the asymptotic growth of Ŝ exhibits a phase transition at q c = 1/2. Specifically, assuming for simplicity that X is centered, then on the one hand for q < 1/2, there is some non-degenerate random variable V such that lim

n→∞ n -1+q Ŝ(n) = V a.s. (2)
In other words, [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF] shows that for q < 1/2, Ŝ has scaling exponent α = 1/(1-q), or equivalently, grows with exponent 1/α, and in particular is super-diffusive since 1/α > 1/2. On the other hand, for q > 1/2, provided that X ∈ L r (P) for some r > 1/(1 -q), the step-reinforced random walk remains diffusive, in the sense that n -1/2 Ŝ(n) converges in law to some Gaussian variable. This phase transition was first established when X is a Rademacher variable, i.e. P(X = 1) = P(X = -1) = 1/2. Indeed, Kürsten [START_REF] Kürsten | Random recursive trees and the elephant random walk[END_REF] observed that Ŝ is then a version of the so-called elephant random walk, a nearest neighbor process with memory which was introduced by Schütz and Trimper [START_REF] Schütz | Elephants can always remember: Exact long-range memory effects in a non-markovian random walk[END_REF] and has then raised much interest. The description of the asymptotic behavior of the elephant random walk has motivated many works, see notably [START_REF] Baur | On a class of random walks with reinforced memory[END_REF][START_REF] Baur | Elephant random walks and their connection to Pólya-type urns[END_REF][START_REF] Bercu | A martingale approach for the elephant random walk[END_REF][START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF][START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF][START_REF] Kubota | Gaussian fluctuation for superdiffusive elephant random walks[END_REF].

Further, when the typical step X has a symmetric stable distribution with index α ∈ (0, 2], Ŝ is the so-called shark random swim, which has been studied in depth by Businger [START_REF] Businger | The shark random swim (Lévy flight with memory)[END_REF]. Its large time asymptotic behavior exhibits a similar phase transition for α > 1, now for the critical parameter q c = 1 -1/α. When α ≤ 1, there is no such phase transition and Ŝ has the same scaling exponent α as S. See also [START_REF] Bertoin | Noise reinforcement for Lévy processes[END_REF] for related results in the setting of Lévy processes.

The results that we just recalled suggest that, more generally, for any steady innovation regime and any typical step X belonging to the domain of normal attraction an α-stable distribution (i.e. such that (1) is fulfilled), then the following should hold. First, for α ∈ (0, 1), the random walk S and its step-reinforced version Ŝ should have the same scaling exponent α = α, independently of the innovation rate. Second, for α ∈ (1, 2], if the innovation rate q is larger than q c = 1 -1/α, then again the scaling exponent of S and Ŝ should coincide, whereas if q < 1 -1/α, then the super-α-diffusive behavior (2) should hold and Ŝ should thus have scaling exponent α = 1/(1 -q) < α. We shall see in Theorems 3.3 and 4.2 that this guess is indeed correct. In particular, the weaker the innovation (or equivalently the stronger the reinforcement), the faster the step-reinforced random walk Ŝ grows. An informal explanation for this phase transition is as follows. When α ∈ (1, 2], the α-diffusive behavior (1) of S relies from some kind of balance between its positive and negative steps (recall that X must be centered, i.e. E(X) = 0). The effect reinforcement of Simon's algorithm for sufficiently small innovation rates q yields certain steps to be repeated much more often than the other, up to the point that this balance is disrupted. More precisely, we shall see that in a steady regime with innovation rate q, the maximal number of repetitions of a same item up to the n-th step of Simon's algorithm grows with exponent 1 -q. For q > q c = 1 -1/α, this is smaller than the growth exponent 1/α of S and repetitions have only a rather limited impact on the asymptotic behavior of Ŝ. At the opposite, for q < q c , some increments have been repeated much more often and the growth of Ŝ is then rather governed by the latter, yielding [START_REF] Baur | On a class of random walks with reinforced memory[END_REF].

We now turn our attention to regimes with slow innovation. Extrapolating from the steady regime, we might expect that reducing the innovation should again speed up the step-reinforced random walk. This intuition turns out to be wrong, and we will see that at the opposite, in slow regimes, diminishing the innovation actually slows down the walk. More precisely, there is another phase transition when α ∈ (0, 1), occurring now for the critical innovation exponent ρ c = α. Specifically, if ρ < α, then we shall see in Theorem 3.1 that Ŝ has always scaling exponent α = 1 (i.e. a ballistic asymptotic behavior which contrasts with the growth with exponent 1/α > 1 for S), whereas for ρ > α, we will see in Theorem 4.1 that Ŝ has rather scaling exponent α = α/ρ > α, that it grows now with exponent 1/α = ρ/α > 1, but nonetheless still significantly slower than S. On the other hand, when α ≥ 1, there is no phase transition for slow innovation regimes and Ŝ has always scaling exponent α = 1. This apparently surprising feature can be explained informally as follows. As it was argued above, for α ∈ (1, 2], E(X) = 0 and the super-diffusive regime (3) results from the disruption of the balance between positive and negative steps when certain steps are repeated much more than others. At the opposite, for α ∈ (0, 1), the typical step X has a heavy tail distribution with E(|X|) = ∞. In this situation, it is well-known that for n 1, |S(n)| has roughly the same size as its largest step up to time n, max{|X i | : 1 ≤ i ≤ n}. Regimes with slow innovation delay the occurence of rare events at which steps are exceptionally large. Therefore they induce a slow down effect for the step-reinforced random walk, up to the point that when the innovation exponent drops below a critical value, Ŝ has merely a ballistic growth. This aspect will be further discussed quantitatively in Section 5.

A somewhat simpler version of the main results of our work are summarized in Figure 1 below. It expresses the scaling exponent α of Ŝ in terms of the scaling exponent α ∈ (0, 2] of S and the innovation parameter ρ > 0. The slow regime corresponds to ρ ∈ (0, 1) and ρ is then the innovation exponent as usual. The steady regime corresponds to ρ > 1, and then the rate of innovation is given by q = 1 -1/ρ. This new parametrization for steady regimes of innovation may seem artificial; nonetheless we stress that the same is actually used for the definition of the one-parameter family of Yule-Simon distributions; see Lemma 2.4.

The cornerstone of our approach is provided by Lemma 2.2, where we observe that the process that counts the number of occurrences of a given item in Simon's algorithm can be turned into a square integrable martingale. The latter is a close relative to another martingale that occurs naturally in the setting of the elephant random walk; see [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF][START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF][START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF][START_REF] Kubota | Gaussian fluctuation for superdiffusive elephant random walks[END_REF], among others. The upshot of Lemma 2.2 is that this yield useful estimates for these numbers of occurrences and their asymptotic behaviors, which hold uniformly for all items.

The plan for the rest of this article is as follows. Section 2 is devoted to preliminaries on the stable central limit theorem, on martingales induced by occurrence counting processes in Simon's algorithm, and on the Yule-Simon distributions. We state and prove our main results in Sections 3 and 4. Finally, several comments are given in Section 5.

Preliminaries

Given two sequence a(n) and b(n) of positive real numbers, it will be convenient to use the following notation throughout this work:

a(n) ∼ b(n) ⇐⇒ lim n→∞ a(n)/b(n) = 1, a(n) ≈ b(n) ⇐⇒ lim n→∞ a(n)/b(n) exists in (0, ∞), a(n) b(n) ⇐⇒ 0 < inf n≥1 a(n)/b(n) ≤ sup n≥1 a(n)/b(n) < ∞.

Background on the stable central limit theorem

We assume in this section that the step distribution belongs to the domain of normal attraction (without centering) of some stable distribution, i.e. that (1) holds for some α ∈ (0, 2]. The Cauchy case α = 1 has some peculiarities and for the sake of simplicity, it will be ruled out from time to time. We present some classical results in this framework that will be useful later on.

We start by recalling that for α = 2, (1) holds if and only if X is centered with finite variance; see Theorem 4 on p. 181 in [START_REF] Gnedenko | Limit distributions for sums of independent random variables. Translated from the Russian, annotated[END_REF]. For α ∈ (0, 1), ( 1) is equivalent to

lim x→∞ x α P(X > x) = c + and lim x→∞ x α P(X < -x) = c -
for some nonnegative constants c + and c -with c + + c -> 0. Finally, for α ∈ (1, 2), (1) holds if and only if the same as above is fulfilled and furthermore X is centered. See Theorem 5 on p. 181-2 in [START_REF] Gnedenko | Limit distributions for sums of independent random variables. Translated from the Russian, annotated[END_REF].

We denote the characteristic function of X by

Φ(θ) = E(exp(iθX)) for θ ∈ R,
and the characteristic exponent of the stable variable Y by ϕ α , that is ϕ α : R → C is the unique continuous function with ϕ α (0) = 0 such that

E(exp(iθY )) = exp(-ϕ α (θ)) for θ ∈ R.
In particular, ϕ α is homogeneous with degree α in the sense that

ϕ α (cθ) = c α ϕ α (θ)
for all c > 0 and θ ∈ R.

In this setting, (1) can be expressed classically as

lim n→∞ Φ(θn -1/α ) n = exp(-ϕ α (θ)), for all θ ∈ R, (3)
but we shall rather use a logarithmic version of (3).

Pick r > 0 sufficiently small so that |1 -Φ(θ)| < 1 whenever |θ| ≤ r, and then define ϕ : [-r, r] → C as the continuous determination of the logarithm of Φ on [-r, r], i.e. the unique continuous function with ϕ(0) = 0 and such that Φ(θ) = exp(-ϕ(θ)) for all θ ∈ [-r, r]. Theorem 2.6.5 in Ibragimov and Linnik [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF] entails that (3) can be rewritten in the form

lim t→∞ tϕ(θt -1/α ) = ϕ α (θ), for all θ ∈ R. (4)
We stress that the parameter t in (4) is real, whereas n in (3) is an integer, and as a consequence, we have also that

ϕ(θ) = O(|θ| α ) as θ → 0.

Martingales in Simon's algorithm

Recall Simon's algorithm from the Introduction, and in particular that σ(n) stands for the number of innovations up to the n-th step. In this work, we will be mostly concerned with the cases where either the sequence σ(•) is regularly varying with exponent ρ ∈ (0, 1), that is

lim n→∞ σ( cn ) σ(n) = c ρ for all c > 0, (5)
or

∞ n=1 n -2 |σ(n) -qn| < ∞ for some q ∈ (0, 1). ( 6
)
It is easily checked that (6) implies σ(n) ∼ qn, and conversally, (6) holds whenever σ(n)/n = q + O(log -β n) for some β > 1. We refer to (5) as the slow regime with innovation exponent ρ ∈ (0, 1), and to (6) as the steady regime with innovation rate q ∈ (0, 1). Often, it is convenient to set ρ = 1/(1 -q) for q ∈ (0, 1) and then view ρ ∈ (0, 1) ∪ (1, ∞) as a parameter for the innovation, with ρ > 1 corresponding to steady regimes.

Several of our results however rely on much weaker assumptions; in any case we shall always assume at least that the total number of innovations is infinite and that the number of repetitions is not sub-linear, i.e.

σ(∞) = ∞ and lim sup n→∞ n -1 σ(n) < 1. (7)
Simon's algorithms induces a natural partition of the set of indices N = {1, 2, . . .} into a sequence of blocks B 1 , B 2 , . . ., where

B j = {k ∈ N : Xk = X j }.
In words, B j is the set of steps of Simon's algorithm at which the j-th item X j is repeated. We consider for every n ∈ N the restriction of the preceding partition to [n] = {1, . . . , n} and write

B j (n) = B j ∩ [n] = {k ∈ [n] : Xk = X j }; plainly B j (n) is nonempty if and only if j ≤ σ(n). Last, we set |B j (n)| = Card B j (n)
for the number of elements of B j (n), and arrive at the following basic expression for the step-reinforced random walk:

Ŝ(n) = ∞ j=1 |B j (n)|X j = σ(n) j=1 |B j (n)|X j , (8)
where the X j are i.i.d. copies of X, and further independent of the random coefficients |B j (n)|.

The identity [START_REF] Bertoin | Universality of noise reinforced Brownian motions[END_REF] incites us to investigate the asymptotic behavior of the coefficients |B j (n)|. In this direction, we introduce the quantities

π(n) = n j=2 1 + 1 -ε j j -1 , n ∈ N, (9)
and the times of innovation

τ (j) = inf{n ∈ N : σ(n) = j} = min B j = inf{n ∈ N : |B j (n)| = 1}.
We start with a simple lemma:

Lemma 2.1. The following assertions hold:

(i) Assume that σ(n) = O(n ρ ) for some ρ < 1. Then π(n) ≈ n.
(ii) Assume (6); then π(n) ≈ n 1-q .

(iii) Assume [START_REF] Bertoin | Noise reinforcement for Lévy processes[END_REF]; then the series

∞ n=1 1/(nπ(n)) converges.
Proof. We have from the definition of π(n) that

π(n) = exp   n j=2 log 1 + 1 -ε j j -1   ≈ exp   n j=2 1 -ε j j -1   .
Next we observe by summation by parts that

n j=2 1 -ε j j -1 = n -σ(n) n -1 + n-1 j=2 j -σ(j) j(j -1) . Assume first σ(n) = O(n ρ ) for some ρ < 1. Then ∞ j=2 σ(j)j -2 < ∞, which yields lim n→∞   n j=2 1 -ε j j -1 -log n   exists in R,
and (i) follows.

Next, when (6) holds, we write

n-1 j=2 j -σ(j) j(j -1) = (1 -q) n-1 j=2 1 j -1 - n-1 j=2 σ(j) -qj j(j -1)
.

The second series in the right-hand side converges absolutely; as a consequence,

lim n→∞   n j=2 1 -ε j j -1 -(1 -q) log n   exists in R,
and (ii) follows.

Finally, assume [START_REF] Bertoin | Noise reinforcement for Lévy processes[END_REF]. There is a < 1 such that σ(k) ≤ ak for all k sufficiently large. It follows that there is some b > 0 such that for all n,

n j=2 1 -ε j j -1 ≥ (1 -a) log n + b.
We conclude that 1/(nπ(n)) = O(n a-2 ), which entails the last claim.

The next result determines the asymptotic behavior of the sequences |B j (•)| for all j ∈ N, and will play therefore a key role in our analysis. Lemma 2.2. Assume [START_REF] Bertoin | Noise reinforcement for Lévy processes[END_REF]. For every j ∈ N, the process started at time τ (j),

π(n) -1 |B j (n)|, n ≥ τ (j),
is a square integrable martingale. We denote its terminal value by

Γ j = lim n→∞ π(n) -1 |B j (n)|,
and have

E(Γ j ) = 1 π(τ (j))
and

Var(Γ j ) ≤ 2 π(τ (j)) ∞ n=τ (j) 1 nπ(n) .
Proof. The martingale property is immediate from Simon's algorithm. More precisely, for any n ≥ τ (j), we have π(n + 1) = π(n) and |B j (n + 1)| = |B j (n)| when ε n+1 = 1 (by innovation), whereas when ε n+1 = 0 , we have π(n + 1) = π(n)(1 + 1/n) and further (by reinforcement)

P(|B j (n + 1)| = |B j (n)| + 1 | F n ) = B j (n)/n and P(|B j (n + 1)| = |B j (n)| | F n ) = 1 -B j (n)/n,
where (F n ) n≥1 denotes the natural filtration of Simon's algorithm. The claimed martingale property follows, and as a consequence, there is the identity

E(|B j (n)|) = π(n)/π(τ (j)) for all n ≥ τ (j). ( 10
)
We next have to check that the mean of the quadratic variation of the martingale

|B j (•)|/π(•) satisfies ∞ n=τ (j) E |B j (n + 1)| π(n + 1) - |B j (n)| π(n) 2 ≤ 2 π(τ (j)) ∞ n=τ (j) 1 nπ(n) ;
thanks to Lemma 2.1, the remaining assertions are then immediate.

In this direction, we first note that the terms in the sum on the left-hand side above that correspond to an innovation (i.e. ε n+1 = 1) are zero and can thus be discarded. Let ε n+1 = 0, so that π(n + 1) = π(n)(1 + 1/n). We then have

E |B j (n + 1)| π(n + 1) - |B j (n)| π(n) 2 ≤ E |B j (n)| + 1 π(n)(1 + 1/n) - |B j (n)| π(n) 2 |B j (n)| n + E |B j (n)| π(n)(1 + 1/n) - |B j (n)| π(n)
On the one hand, since

|B j (n)| + 1 π(n)(1 + 1/n) - |B j (n)| π(n) = 1 -|B j (n)|/n π(n)(1 + 1/n) ∈ 0, 1 π(n) ,
we deduce from [START_REF] Bornholdt | World Wide Web scaling exponent from Simon's 1955 model[END_REF] the bound

E |B j (n)| + 1 π(n)(1 + 1/n) - |B j (n)| π(n) 2 |B j (n)| n ≤ 1 nπ(n)π(τ (j))
.

On the other hand, since

|B j (n)| π(n)(1 + 1/n) - |B j (n)| π(n) 2 ≤ |B j (n)| 2 π(n) 2 n 2 ≤ |B j (n)| π(n) 2 n ,
using again (10), we get

E |B j (n)| π(n)(1 + 1/n) - |B j (n)| π(n) 2 ≤ 1 nπ(n)π(τ (j))
.

The proof of the statement is now complete.

As an immediate consequence, we point at the following handier estimate for the second moment of Γ j .

Corollary 2.3. Assume [START_REF] Bertoin | Noise reinforcement for Lévy processes[END_REF] and further that π(n) n a for some a > 0. Then

E(Γ 2 j ) 1 τ (j) 2a .
Proof. On the one hand, there is the lower bound E(Γ 2 j ) ≥ E(Γ j ) 2 . On the other hand, our assumption also entails for some b, b > 0, we have

n≥ 1/(nπ(n)) ≤ b n≥ n -1-a ≤ b -a ,
and we conclude with Lemma 2.2.

Yule-Simon distributions

Recall that the slow and the steady regimes have been defined by ( 5) and ( 6), respectively. Simon [START_REF] Simon | On a class of skew distribution functions[END_REF] observed that in each regime, the empirical measure of the sizes of the blocks |B j (n)| converges to a deterministic distribution. Lemma 2.4. [Simon [START_REF] Simon | On a class of skew distribution functions[END_REF]] Let ρ > 0. For 0 < ρ < 1, consider the regime (5) of slow innovation with exponent ρ, whereas for ρ > 1, set q = 1 -1/ρ ∈ (0, 1) and consider the regime (5) of slow innovation with exponent ρ. In both regimes, for every k ∈ N, we have

lim n→∞ 1 σ(n) Card{j ≤ σ(n) : |B j (n)| = k} = ρB(k, ρ + 1),
where B is the Beta function and the convergence holds in L p for any p ≥ 1.

The limiting distribution in the statement is called the Yule-Simon distribution with parameter ρ. Strictly speaking, Simon only established the stated converge in expectation. A classical argument of propagation of chaos yields the stronger convergence probability; see e.g. Section 5 in [START_REF] Baur | On a two-parameter Yule-Simon distribution[END_REF], and since the random variables in the statement are obviously bounded by 1, convergence in L p also holds for any p ≥ 1.

The next lemma will be needed to check some uniform integrability properties.

Lemma 2.5. Let 0 < β ≤ ρ and assume either (i) or (ii) is fulfilled, where:

(i) ρ ∈ (0, 1) and the slow regime (5) holds with exponent, (ii) ρ > 1 and the steady regime (6) holds with rate q = 1 -1/ρ.

Then sup n≥1 1 σ(n) σ(n) j=1 E(|B j (n)| β ) < ∞. Remark 2.6. Since B(k, ρ + 1) ∼ Γ(ρ + 1)k -(ρ+1) as k → ∞, we have that ∞ k=1 k β ρB(k, ρ + 1) < ∞
for any β < ρ, in agreement with Fatou's lemma and Lemmas 2.4 and 2.5.

Proof. (i) Recall from Lemma 2.1 that in the slow regime, there are the bounds n/c ≤ π(n) ≤ cn for all n ∈ N, where c > 1 is some constant. Since, from Lemma 2.2,

E(|B j (n)|) = π(n)/π(τ (j)) ≤ c 2 n/τ (j),
we get by Jensen inequality that

σ(n) j=1 E(|B j (n)| β ) ≤ c 2β n β σ(n) j=1 τ (j) -β = O n β σ(n)(τ (σ(n))) -β ,
where for the O upperbound, we used the fact that the inverse function τ of σ is regularly varying with exponent 1/ρ (Theorem 1.5.12 in [START_REF] Bingham | Encyclopedia of Mathematics and its Applications[END_REF]), and Proposition 1.5.8 in [START_REF] Bingham | Encyclopedia of Mathematics and its Applications[END_REF] since -β/ρ > -1. On the other hand, since τ is the right-inverse of σ, we have τ (σ(n)) ≤ n ≤ τ (σ(n) + 1), so again by regular variation, τ (σ(n)) ∼ n. Finally

σ(n) j=1 E(|B j (n)| β ) = O(σ(n)),
as we wanted to verify.

(ii) The proof is similar to (i), using now that there exists c > 0 such that

E(|B j (n)| 2 ) ≤ c(n/j) 2-2q
for all j ∈ N and n ≥ τ (j), as it is readily seen from Corollary 2.3.

Strong limit theorems

In this section, we will establish two strong limit theorems for step-reinforced random walks, the first concerns slow innovation regimes, and the second steady ones.

Ballistic behavior

Theorem 3.1. Suppose that

σ(n) = O(n ρ ) as n → ∞,
for some ρ ∈ (0, 1), and that

P(|X| > x) = O(x -β ) as x → ∞,
for some β > ρ. Then

lim n→∞ n -1 Ŝ(n) = V a.s.
where V is some non-degenerate random variable.

We will deduce Theorem 3.1 by specializing the following more general result.

Lemma 3.2. Assume (7) and set

Γ * j = sup n≥τ (j) |B j (n)|/π(n), j ∈ N. Provided that ∞ j=1 Γ * j |X j | < ∞ a.s., (11)
we have

lim n→∞ Ŝ(n)/π(n) = V a.s., with V = ∞ j=1 Γ j X j .
Proof. Thanks to [START_REF] Businger | The shark random swim (Lévy flight with memory)[END_REF], the claim follows from (8) and Lemma 2.2 by dominated convergence.

Proof of Theorem 3.1. Recall from Lemma 2.1(i) that π(n) ≈ n. From Lemma 3.2, it thus suffices to check that

∞ j=1 E (Γ * j |X j |) ∧ 1) < ∞, (12)
since then, the condition (11) follows.

Without loss of generality, we may assume that β < 1. Pick a > 0 sufficiently large so that σ(n) ≤ an ρ for all n ≥ 1 and

P(|X| > x) = ax -β for all x > 0.
Since X j is a copy of X which is independent of Γ * j , we have

E (Γ * j |X j |) ∧ 1 = 1 0 P(Γ * j |X j | > x)dx ≤ a(1 -β) -1 E((Γ * j ) β ).
Recall from Lemma 2.2 that |B j (•)|/π(•) is a closed martingale with terminal value Γ j . Then by Doob's maximal inequality, there is some numerical constant c β > 0 such that E((Γ * j ) β )) ≤ c β E(Γ j ) β , and hence again from Lemma 2.2,

E (Γ * j |X j |) ∧ 1 = O(τ (j) -β ).
Finally, since τ (j) ≥ (j/a) 1/ρ , we conclude that

E (Γ * j |X j |) ∧ 1 = O(j -β/ρ ) as j → ∞,
which ensures (12).

Super-α-diffusive behavior

We next turn our attention to the steady regime.

Theorem 3.3. Suppose (6) holds with q < 1/2 and that

E(|X| β ) < ∞ and E(X) = 0,
for some β > 1/(1 -q). Then

lim n→∞ n q-1 Ŝ(n) = V in L β (P)
where V is some non-degenerate random variable.

The proof of Theorem 3.3 relies on the following martingale convergence result.

Lemma 3.4. Assume (7) and let β ∈ [START_REF] Aaronson | Characteristic functions of random variables attracted to 1-stable laws[END_REF][START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF]. Suppose that X ∈ L β (P) with E(X) = 0, and further that

∞ j=1 E(Γ β j ) < ∞. (13)
The process

V n = n j=1 Γ j X j , n ∈ N
is then a martingale bounded in L β (P); we write V ∞ for its terminal value. We have

lim n→∞ Ŝ(n)/π(n) = V ∞ in L β (P).
Proof. The assertion that the process V n is a martingale is straightforward since the variables X j are i.i.d., centered, and independent of the Γ j . The assertion of boundedness in L β (P) then follows from the assumption ( 13), the Burkholder-Davis-Gundy inequality, and the fact that, for any sequence (y j ) j∈N of nonnegative real numbers, since

β ≤ 2,   ∞ j=1 y 2 j   β/2 ≤ ∞ j=1 y β j .
The convergence of Ŝ(n)/π(n) is proven similarly. Specifically, we observe from (8) that

V σ(n) -Ŝ(n)/π(n) = σ(n) j=1 (Γ j -|B j (n)|/π(n)) X j ,
and recall that the variables X j are independent of those appearing in Simon's algorithm. By the Burkholder-Davis-Gundy inequality, there exists a constant c β ∈ (0, ∞) such that

E V σ(n) -Ŝ(n)/π(n) β ≤ c β E(|X| β ) σ(n) j=1 E |Γ j -|B j (n)|/π(n)| β .
We know from Lemma 2.2 that for each j ≥ 1,

lim n→∞ E |Γ j -|B j (n)|/π(n)| β = 0,
and further by Jensen's inequality, that

E |Γ j -|B j (n)|/π(n)| β ≤ 2 β E Γ β j .
The assumption [START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF] enables us to complete the proof by dominated convergence.

Proof of Theorem 3.3. It follows from ( 6) and Lemma 2.1(ii) that

τ (n) ∼ n/q and π(n) n 1-q , (14)
and then from Corollary 2.3 that E(Γ 2 j ) j -2+2q . Without loss of generality, we may suppose β ≤ 2. Hence, by Jensen's inequality, we have

∞ j=1 E(Γ β j ) ≤ ∞ j=1 E(Γ 2 j ) β/2 < ∞.
An appeal to Lemma 3.4 completes the proof.

Weak limit theorems

In this section, we will establish two weak limit theorems for step-reinforced random walks, depending on the innovation regimes.

Super-ballistic behavior

Theorem 4.1. Suppose that X belongs to the domain of normal attraction of a stable law (i.e. (1) holds) with index α ∈ (0, 1), and that (5) holds for some ρ ∈ (α, 1). Then

lim n→∞ σ(n) -1/α Ŝ(n) = Y in law
where Y is an α-stable random variable.

Under the assumptions of Theorem 4.1, the step-reinforced random walk grows roughly like n ρ/α , and since 1 < ρ/α < 1/α, its asymptotic behavior is both super-ballistic and sub-α-diffusive.

Proof. Note first that, since ρ > α, nσ(n) -1/α goes to 0 as n → ∞, and a fortiori so does |B j (n)|σ(n) -1/α uniformly for all j ∈ N. We fix θ ∈ R and get from (8) that for n sufficiently large

E(exp(iθσ(n) -1/α Ŝ(n))) = E   exp   - σ(n) j=1 ϕ(θσ(n) -1/α |B j (n)|)    
We focus on the sum in the right-hand side, and first consider the terms with

|B j (n)| ≤ k for some fixed k ∈ N. Write j:|Bj (n)|≤k ϕ(θσ(n) -1/α |B j (n)|) = 1 σ(n) k =1 ϕ(θσ(n) -1/α )σ(n)N (n), where N (n) = Card{j ≤ σ(n) : |B j (n)| = }. Next, recall from (4) that as n → ∞, ϕ(θσ(n) -1/α )σ(n) ∼ ϕ α (θ ) = ϕ α (θ) α .
We now deduce from Lemma 2.5 that for any fixed k ∈ N, there is the convergence lim

n→∞ j:|Bj (n)|≤k ϕ(θσ(n) -1/α |B j (n)|) = ϕ α (θ) k =1 α ρB( , ρ + 1) in L p (P)
for every p ≥ 1.

We can next complete the proof by an argument of uniform integrability. Recall that ϕ(λ) = O(|λ| α ) as λ → 0 and pick β ∈ (α, ρ). There exists a > 0 such that for all n sufficiently large and all k ≥ 1, there is the upper bound

j:|Bj (n)|>k ϕ(θσ(n) -1/α |B j (n)|) ≤ a k α-β σ(n) ∞ j=1 |B j (n)| β ,
and the same inequality holds with ϕ α replacing ϕ. We can then deduce from the preceding paragraph in combination with Lemma 2.5 that actually

lim n→∞ σ(n) j=1 ϕ(θσ(n) -1/α |B j (n)|) = ϕ α (θ) ∞ =1 α ρB( , ρ + 1) in probability.
It now suffices to recall that ϕ ≥ 0, so by dominated convergence,

lim n→∞ E(exp(iθσ(n) -1/α Ŝ(n))) = exp -ϕ α (θ) ∞ =1 k α ρB(k, ρ + 1) ,
which completes the proof.

α-diffusive behavior

Theorem 4.2. Suppose that X belongs to the domain of normal attraction without centering of a stable law (i.e. (1) holds) with index α ∈ (0, 2], and that (6) holds for some q ∈ (0, 1). Suppose further that q > 1 -1/α when α > 1.

Then

lim n→∞ n -1/α Ŝ(n) = Y in law
where Y is an α-stable random variable.

The proof of Theorem 4.2 requires the following uniform bounds Lemma 4.3. Suppose (6) holds for some q ∈ (0, 1) and take any β ∈ (0, 1/(1q)). Then lim

n→∞ sup j≥1 |B j (n)|n -1/β = 0 in probability.
Proof. The claim is obvious when β < 1, so we focus on the case β ≥ 1. In this direction, recall from Lemma 2.2 that |B j (n)|/π(n) is a square integrable martingale with terminal value Γ j . Recall also from Lemma 2.1(ii) and Corollary 2.3, that in the regime (6), π(n) ≈ n 1-q and E(Γ 2 j ) j 2q-2 . There is thus some constant a > 0, such that for any η > 0 arbitrarily small, we have

P(|B j (n)| > ηn 1/β ) ≤ aη -2 n 2-2q-2/β j 2q-2 . ( 15
)
Suppose first that q < 1/2, so j≥1 j 2q-2 < ∞ and therefore

∞ j=1 P(|B j (n)| > ηn 1/β ) = O(n 2-2q-2/β ).
Since 1 -q < 1/β, our claim follows.

Then suppose that q = 1/2; using j≤n j -1 ∼ log n and |B j (n)| = 0 for j > n, we get

∞ j=1 P(|B j (n)| > ηn 1/β ) = O(n 1-2/β log n).
Since 1/β > 1/2, our assertion is verified.

Finally, suppose that q > 1/2; using j≤n j 2q-2 ≈ n 2q-1 and |B j (n)| = 0 for j > n, we get

∞ j=1 P(|B j (n)| > ηn 1/β ) = O(n 1-2/β ).
Since again 1/β > 1/2, the proof is complete. Lemma 4.3 enables us to duplicate the argument for the proof of Theorem 4.1, as the reader will readily check.

Miscellaneous remarks

• Technically, the fact that the indices of the steps at which innovations occur are deterministic eases our approach by pointing right from the start at the relevant quantities. Although our statements are only given for deterministic sequences (ε j ), they also apply to random sequences (ε j ) independent of (X j ), provided of course that we can check that the requirements hold a.s. A basic example, which has been chiefly dealt with in the literature, is when the ε j are i.i.d. samples of the Bernoulli distribution with parameter q ∈ (0, 1), as then (6) obviously holds a.s. Plainly independence of the ε j is not a necessary assumption, and much less restrictive correlation structures suffice. For instance, if we merely suppose that each ε j has the Bernoulli law with parameter q j such that n≥2 n -2 | n j=2 (q j -q)| < ∞, and that |Cov(ε j , ε )| ≤ |j -| -a for some a > 0, then one readily verifies that ( 6) is fulfilled a.s. Similar examples can be developed to get slow innovation regimes, for instance assuming that each variable ε j has a Bernoulli law with q(j) ≈ j ρ-1 and again a mild condition on the correlation.

• Dwelling on an informal comment made in the Introduction, it may be interesting to compare the step-reinforced random walk Ŝ(n) with its maximal step X * n = max 1≤j≤n | Xj |. Assume α ∈ (0, 2), and that P(|X| > x) ≈ x -α (recall Section 2.1 about characterization of stable domaines of normal attraction). Plainly, there is the identity X * n = X * σ(n) , where X * n = max 1≤j≤n |X j |, from which we deduce that σ(n) -1/α X * n converges in distribution as n → ∞ to some Frechet variable. Comparing with the results in Sections 3 and 4, we now see that in the slow regime with innovation exponent ρ ∈ (0, 1), Ŝ grows with the same exponent as X * when α > ρ, and with a strictly larger exponent if α < ρ. Similarly, in the steady regime with innovation rate q ∈ (0, 1), Ŝ grows with the same exponent as X * when α > ρ = 1/(1-q) and with a strictly larger exponent if α < ρ. In other words, the maximal step X * has a sensible impact in the strong limit theorems of Section 3, but its role is negligible for the weak limit theorems of Section 4.

• We have worked in the real setting for the sake of simplicity only; the arguments work as well for random walks in R d with d ≥ 2. In this direction, one notably needs a multidimensional version of (4), which can be found in Section 2 of Aaronson and Denker [START_REF] Aaronson | Characteristic functions of random variables attracted to 1-stable laws[END_REF]. The same sake of simplicity (possibly combined with the author's lazyness) motivated our choice of working with domains of normal attraction rather than with domains of attraction. Most likely, dealing with this more general setting would only require very minor modifications of the present arguments and results.

• It would be interesting to complete the strong limit results (Theorems 3.1 and 3.3) and investigate the fluctuations n -1/ α Ŝ(n) -V as n → ∞. In the setting of the elephant random walk, Kubota and Takei [START_REF] Kubota | Gaussian fluctuation for superdiffusive elephant random walks[END_REF] have recently established that these fluctuations are Gaussian.

• The case where the generic step X has the standard Cauchy distribution is remarkable, due to the feature that for any a, b > 0, aX 1 + bX 2 has the same distribution as (a + b)X, where X 1 and X 2 are two independent copies of X. It follows that n -1 Ŝ(n) has the standard Cauchy distribution for all n, independently of the choice of the sequence (ε j ). This agrees of course with Theorems 3.1 and 4.2.

Figure 1 :

 1 Figure 1: Scaling exponent α of a step-reinforced random walk in terms of the innovation parameter ρ and the scaling exponent α of the original random walk. Results above the diagonal ρ = α are strong (convergence in probability), those below the diagonal are weak (convergence in distribution).

The precise definition of these regimes will be given later on.


"What difference does it make?" Studying urban popular music from before the generalization of the gramophone: the example of the First World war repertoire.

John Mullen, Université de Rouen Normandie, Research team ERIAC For the last ten years I have been working on popular song, mostly in Britain, between 1880 and 1918, and often concentrating on the songs and the music hall industry of the First World War. 1 Taking up this work, I found it to be wedged between two different traditions. There existed a number of analytical books on the Victorian music hall 2 , but very few on the twentieth century, and on the other hand there was a tremendous amount of work being done on popular music produced after 1950, rock music and later forms.

One of my objectives was to explore to what extent the work on the Victorian age had produced conclusions which could still be applied to the repertoires and processes of the early twentieth century, and, moving in the other direction, what analyses and perspectives on late twentieth century and early twenty first century popular music might already apply to the products and industries of a century ago.

Working specifically on the Popular song of the first world war, I also had to fit my work in with the field of First World War history. This could be delicate. Like other researchers in Popular music, I had to justify my object of study: in a period of total war, in which deaths and injuries were counted in millions, it might be considered trivial to study products of entertainment. Such pressure was reinforced by the sacralization and the politicization of Great War memory in a number of countries, and particularly in Britain, where the centenary commemorations have been the occasion for a full-scale political debate. 3 The study of popular song in order to help understand the worries, fantasies and priorities of ordinary people might it not threaten an established national narrative of unquestioning mass jingoism?

In this No-man's land between First World War history and Popular Music Studies, two questions in particular seemed important to me Firstly, what was the specificity of a popular song as an object of historical study: what difference did it make that one was studying a song rather than another object or process? And secondly what was the specificity of songs from a century ago as objects of Popular Music studies: what difference does it make that these songs were hits before the generalization of the gramophone? 4 3 John Mullen, "Experiences and Contradictions: How the British Celebrated the Centenary of 1914", Revue française de civilisation britannique, XXI-1, 2015. 4 Many readers will be unfamiliar with the songs I am speaking of. These three available on Youtube can serve to give a very general idea of the sounds involved : Marie Lloyd's "A Little of What You Fancy Does You Good" https://www.youtube.com/watch?v=X6P_zU5oxF8 ; "Goodbyee" by Courtland and Jeffries https://www.youtube.com/watch?v=jVavtX80m3Q ; "Keep the Home Fires Burning" John McCormack https://www.youtube.com/watch?v=5P8UokgVqWs

First World War History

To deal with the first of these questions, I first need to explain a little about the historiography of the First World War, a subject which has produced tens of thousands of books. Jay Winter and Antoine Prost in their book 5 , describe the historiography of the war as being marked by three main types of history, each dominating by turn academic and publishing interest. Firstly, military and diplomatic history ruled the roost for thirty years after 1918. Secondly, in the post-war boom, social history rose to influence, finally, from the 1980s on, cultural history (somewhat more difficult to define) was the key player.

The types of history can mostly be distinguished by the kinds of questions they considered were a priority. Diplomatic and military history was in its early decades very much "history from above". Its key questions turned around leadership strategies of governments and general and how they might have been improved. The questions of how the war was won, and who was to blame for the outbreak of the war, were much explored. The aim of justifying policy was often present. There was little interest in ordinary people or ordinary soldiers. The lived experience on the ground was simply not considered to be a suitable subject of historical study, to such an extent that the great French historian, Pierre Renouvin, seriously wounded in the war, never mentions his own experience in his historical works.

Later military history, and there continue to be over a hundred books of military history of the First World War published every year, put, on the contrary, much emphasis on the lived experience of the individual soldier, nurse, priest, machine gunner, officer, prisoner of war etc. This is the type of book which far outsells any other sort.

After the Second World War, the multiplication in the number of universities, the further progress of the democratic spirit in society, the decline of deference, the possibility given to some working-class youngsters to become historians and the growing influence of Marxism in university circles led to the rise of social history. New questions were asked about the war, questions about the transformation of social structures, the changes in women's lives, the effect on civilians. Books would be published about women munition workers or footballers, about the life of political parties, voluntary organizations or churches.

Finally, twenty years later, the third type of history will come to dominate First World War studies. This is cultural history. Vast (indeed the number of historians was still growing rapidly) and difficult to define precisely, cultural history shared with social history an interest for the ordinary folk. Its rise to pre-eminence reflected a reduction in the influence in Marxism in academic circles, concomitant with the collapse of the Soviet bloc, but also a rise in interdisciplinary approaches which borrowed from many other scholars who were not historians (cultural studies people, literary analysts and the like). Cultural history's strongest point was no doubt its capacity to massively extend the range of what could be considered as an object of historical study. Its weakest point was a tendency to minimize the role of the state and of collective organization in the course of history and the forging of individual consciousness.

The study of the constitution and uses of collective memory make up a considerable part of cultural history concerning the First World War, and more generally, the history of representations has been on the rise.

The rules of studying cultural production

But also within cultural history we find the study of cultural expression in wartime, be it theatre, literature, postcards, war monuments, soldiers letters, letters to the press etc. And, fortunately for my career, ten years ago, no one had seriously studied the history of British popular song in the First World War.

Studies of these cultural expressions have some characteristics in common. They concern themselves with what might be called the grammar of their objects of study. For example, the studies of the thousands of commemorative monuments to the war dead needed to know who financed the building of the monuments, who made the key decisions, and so on. Researchers found, for example, that families of dead or wounded had little say in the decisions about the content of monuments: local elites took over these policy decisions. The researchers went on to analyse the repertoire of visual symbols, texts, ceremonies etc., used for the establishment of the monuments.

To take another example, those who studied postcards produced during the war on war themes (often comic or sentimental) collected information about the situation and priorities of the producers (general very small businesses). The closeness of the producers to their customers meant that the images they felt would sell reflected the opinions, worries and fantasies of the millions of ordinary people who bought them.

Is it the same for songs? It would seem, then, that there are many points in common between these studies of means of cultural expression and the study of commercial wartime songs. These songs had a similar need to follow closely developments in fantasies, anxieties and opinions of their mass audience, at the same time as avoiding repeating excessively previous productions.

To understand the song repertoires and the music industry processes of the First World War it was necessary to identify the different actors (theatre chains, touring artistes, publishing companies, freelance lyricists), the legal and economic framework (no royalties for lyricists, legal dangers attached to 'vulgar' songs), traditional and newer forms of musical spectacle and their constraints (music hall, revue, pantomime, musical comedy), ideological parameters (patriotism, respectability or expression of working class identity) and so on.

An important difference concerning the study of song resided no doubt in the tremendous amount of emotion and identity tied up with musical repertoires. Whether it be in 1916 or 2016, there are few people who are as attached to monuments or postcards as they are to their favourite songs and performances of songs.

Popular Music Studies and the First World War repertoire

Coming at the same question -"What difference does it make?" from the contrary point of view, I will now evaluate what methods and approaches current in Popular Music Studies today can be fruitfully applied to the First World War repertoires and processes. Bennet, Shank and Toynbee's influential Popular Music Studies Reader helpfully divides the field into the following nine sections.

-Music as sound, music as text -Making music -Subculture scenes and tribes -Popular music and everyday life -Musical diasporas -Music industry -Popular music and technology -Popular music media -Popular music, gender and sexuality How can each of these apply to the music a hundred years ago? There is not space in this paper to discuss all of them, but let us explore some.

Musical material and the music industry Certainly, the music can be analysed as sound and text. For the texts, if the British wartime repertoire is generally not complex (partly because lyricists sold songs for a small one-off payment and didn't receive royalties), that of some other countries such as France is sophisticated in theme and lyric, exploring a wide variety of problems and dreams of wartime experience (infidelity, disfigurement, hatred of the enemy, national morality etc.) And any repertoire, including the British, can be studied for its ideological implications (hits included a few racist and anti-Semitic songs, many songs about respectability and dozens expressing worry about how women's roles had changed, for example).

The music of the British repertoire cannot be innovative. The artiste is generally on tour and the local house orchestra must learn, on Monday morning, the artiste's songs for the first performance Monday evening. Innovative or difficult music would likely result in an artiste not being hired. In addition, there was of course no studio work to analyse as there is today. The low level of musical control and musical ambition would make it difficult to carry out analyses of the sort Susan Fast did on Led Zeppelin's classic hit. Some elements of musicological analysis proper have nevertheless supplemented political and social analyses of First World War songs in some publications. 6 Such analyses are not without their difficulties, in that the technical language of music is not generally available to readerships of First World War history books, be they academic of wider. Since the historian, in addition to being a researcher, is also a storyteller, this can be problematic.

There was one element of sound content, the most intimate element, over which the artiste retained much control: the voice. I have been researching on the use of voice in the music-hall. The lack of microphones in a hall of two or three thousand seats obviously limited the repertoire of voices that could be employed. The choice of voice tended to be made on a theatrical rather than a technical basis, to project a character the audience can love or laugh at or both. The choice of a voice slightly the worse for drink, 7 a voice of operetta, 8 a cockney working class voice, 9 or a gendered voice 10 (for the many cross-dressing artistes) allows the study of the ideological and social underpinnings of these choices.

Music-making as an industry is certainly a very fruitful domain to examine a hundred years ago. The industry was being transformed in Britain at this time by the concentration of capital, which led to the domination of chains of theatres which could maintain profits by making economies of scale, and by signing exclusive contracts with the bigger stars, denying access to talent for independent local theatres. These chains were quoted on the Stock Exchange and could pay out hefty dividends to their shareholders. Many local theatres went bankrupt or were bullied into joining the chains. Regular crises of overproduction occurred in the industry as local markets for shows were flooded by newly opened theatres, before some went bankrupt or were converted into cinemas.

Pre-industrial traditions, based on the exchange of goodwill and gifts, were slowly disappearing, but the position of the music hall owner or manager as someone only recently accepted in local elites ensured the continuance of some of these traditions (free use of the theatre for charity shows was almost obligatory, for example). The rise of trade unionism, in particular after the music hall strike of 1907 in London, had very much influenced the industry, establishing minimum wage rated for orchestra players, and an arbitration board in collaboration with the government to mediate in industrial disputes. The union journal The Performer allows the researcher a privileged insider's view. 11 The Music-hall artistes' trade union, the Variety Artistes Federation, considered itself very much part of the wider trade union movement, and its paper carried for many years on its front page the motto "the greatest enemy to freedom is not the tyrant, but the contented slave".

One of the most interesting areas to research is to try to tease out the links between economic developments in the industry and artistic content, which mutually influence each other. The rise of the revue, a single all-evening show, threatened, after 1911, the older music hall evening made up of individual and constantly changing turns. The revue benefitted from economies of scale and made it possible to hire dancing troupes and other new groups of skilled workers. The revue could then tour nationally or even internationally as a unit. The revue was a result of the concentration of capital, and it integrated many music hall stars and content. But it also transformed the content. Centralized artistic control allowed more considered experimentation with the building up of atmosphere, for example, which had been all but impossible for the music hall singer who had his or her fifteen minutes after the elephant act and before the acrobats. One of the results was the huge rise in romantic content in the revue compared with the music hall.

Questions of identity

A priori much less productive for our period is the approach based on "Subculture, scenes and tribes". The teenage market did not and could not exist, leisure time and education were very much less developed. The choice of music for working class people often boiled down to going to the music-hall, attending the cheap classical concerts at the town hall, or being satisfied with listening to street singers or sitting at home "When Uncle sings the only song he knows"12 . The popular music market was far less segmented than today: twelve year olds and sixty year olds went to the same shows. And the infrastructure and leisure time for fan clubs, fanzines, or Facebook groups around musical tastes was not available.

Nevertheless, the use of musical production to express and reinforce identities is well worth looking at a century ago. At the time, it was mostly a question of class identities. The music hall stars sang in theatres whose furnishing imitated the homes of the bourgeoisie, but, for the most part, sang in working class accents about everyday working class life. Numerous songs and practice try to situate working class identity in relation to such ideological parameters as respectability or vulgarity. Other identities counted too: attitudes to the colonized Other in the racist songs of the music-hall deserve study, as do the contradictory attitudes to Irishness in the repertoire.

As for the domain of "popular music, gender and sexuality", gender identities are already omnipresent. Many dozens of songs of the British wartime repertoire express anxiety concerning the changing roles of women. The songs do not usually either denounce or celebrate the new roles, but express ambiguous attitudes: it's wonderful and dangerous, a point of view which was certainly very prevalent among both women and men at the time. The French wartime repertoire is characterized by an explorationoften through melodrama-of fears around gender roles and couple relationships. Songs about the soldier who receives a letter discovering his wife is unfaithful, about the soldier returning on leave after twelve months to find a baby in the house, about women replacing men as head of the household after the war, or about the worries of a disfigured soldier about the loyalty of his fiancée, are typical examples. 13 "Popular music media" and "technology and popular music" are two themes which would be much less fruitful for the historical research we are dealing with. Critical discourse about popular music was very little developed. Specialized magazines such as Phono Record contain fascinating anecdotes such as the shock people at a funeral experienced on finding that the deceased had recorded himself singing a hymn to be played at his own funeral, but they were mostly about the industry or about technical questions. Reviews of shows in the national or local press tended to be uniformly positive, and the expert audience required to sell sophisticated critiques of popular music did not exist.

What difference does it make? What, then, is different about working on songs from a century ago. Firstly, the repertoire, though large, is smaller: one can more easily construct a representative corpus of five hundred songs from 1916 than from 2016.

Secondly, there is the question of archives. When one works, as I am doing at the moment, on the 1970s, the problem is that there is far too much archival material, recordings, videos, people to interview, press coverage and so on. For the 1900s, much has been lost. In some countries, like France, the existence at the time of an official censorship of songs, whose offices kept a copy of every song examined as well as the decision taken, make life easy for the historian. In other countries, most of the repertoire has disappeared forever, or can only be reconstituted by having access to plenty of time and money. The British case is intermediate: there are 1200 songs from the war years in the British Library and at least 300 recordings of songs are available. Nevertheless, many British songs have been lost. This brief paper has, I hope, clarified some of the main issues in historical popular music studies. In conclusion, I would like to open up the question of comparative work. What would be the usefulness of a comparative study of the top twenty most popular songs from 1918 and from

See in particular Christina Gier Singing Soldiering and Sheet Music in America during the First World War, Lexington, 2016.

See John Mullen, "Red Plush and Picket Lines: Music-Hall Trade Unionism 1906-1920" in Rosemary Golding (Ed), The Music Profession in Britain 1780-1920, London, Routledge, forthcoming

The title of a 1904 music-hall song.

Anne Simon, Chanter la Grande Guerre: les Poilus et les Femmes, Paris, Champ Vallon, 2014.

2018 in a given country? It might shed light on what is relatively permanent (the three-minute song, the singalong chorus, the role of virtuosity, the centrality of the transgressive) and what is ephemeral in this mass activity which interests us so much.

John Mullen

John Mullen is Professor of British Studies at the University of Rouen in France. He has written widely on the history of British popular music: on music hall, on music festivals and on rock and politics. His book The Show Must Go On: Popular Song in Britain during the First World War was published in the Ashgate Popular and Folk Music series in 2015.