Cette thèse porte sur l'étude de la concentration autour de la moyenne de fonctions de variables aléatoires indépendantes à l'aide de techniques de martingales et d'inégalités de comparaison.Dans une première partie, nous prouvons des inégalités de comparaison pour des fonctions générales séparément convexes de variables aléatoires indépendantes non nécessairement bornées. Ces résultats sont établis à partir de nouvelles inégalités de comparaison dans des classes de fonctions convexes (contenant, en particulier, les fonctions exponentielles croissantes) pour des variables aléatoires réelles uniquement dominées stochastiquement.Dans la seconde partie, nous nous intéressons aux suprema de processus empiriques associés à des observations i.i.d. Le point clé de cette partie est un résultat d'échangeabilité des variables. Nous montrons d'abord des inégalités de type Fuk-Nagaev avec constantes explicites lorsque les fonctions de la classe ne sont pas bornées. Ensuite, nous prouvons de nouvelles inégalités de déviation avec une meilleure fonction de taux dans les bandes de grandes déviations dans le cas des classes de fonctions uniformément bornées. Nous donnons également des inégalités de comparaison de moments généralisés dans les cas uniformément borné et uniformément majoré. Enfin, les résultats de la première partie nous permettent d'obtenir une inégalité de concentration lorsque les fonctions de la classe ont une variance infinie. / This thesis deals with concentration properties around the mean of functions of independent random variables using martingale techniques and comparison inequalities.In the first part, we prove comparison inequalities for general separately convex functions of independent and non necessarily bounded random variables. These results are based on new comparison inequalities in convex classes of functions (including, in particular, the increasing exponential functions) for real-valued random variables which are only stochastically dominated.In the second part, we are interested in suprema of empirical processes associated to i.i.d. random variables. The key point of this part is a result of exchangeability of variables. We first give Fuk-Nagaev type inequalities with explicit constants when the functions of the considered class are unbounded. Next, we provide new deviation inequalities with an improved rate function in the large deviations bandwidth in the case of classes of uniformly bounded functions. We also provide generalized moment comparison inequalities in uniformly bounded and uniformly bounded from above cases. Finally, results from the first part allow us to prove a concentration inequality when the functions of the class have an infinite variance.

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This thesis began (in thought) as a response to the killing of Trayvon Martin in 2012 and that of Mike Brown not too long after, and the many victims who succumbed to some form of racial profiling of another before these deaths, in-between, and after. Desmond Cole wrote an article in 2015 that further precipitated the thought into action and the desire to address racial profiling in writing form. In the thesis I take a philosophical approach to racial profiling, and although in the first two chapters I address the ordinary discussions surrounding racial profiling, in the latter two I tackle the problem of moral responsibility which I take to be central. In the first part of the thesis I defend the policy in the case of illegal weapons possession based on Henry Shue’s principle of basic rights, but in the latter part I question this assertion. Even if blacks were shown to commit more of certain crimes or even violent crimes, that does not address the fact that crime arises out of context and in the case of “black crime” out of a racialized context. In the latter part of the thesis I work through the problem of collective and personal moral responsibility, eventually maintaining that not only is reparations just, but for racial profiling to be justified investment must be made into racialized communities with high rates of poverty. This is because collective responsibility must be taken for the societal oppression and discrimination that has partly resulted in high rates of racialized crime. / Thesis / Master of Arts (MA)

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Chapter 1 introduces Jensen Inequality and its geometric interpretation. Some useful criteria for checking the convexity of functions are discussed. Many applications in various fields are also included. Chapter 2 deals with Schur Inequality, which can easily solve some problems involved symmetric inequality in three variables. The relationship between Schur Inequality and the roots and the coefficients of a cubic equation is also investigated. Chapter 3 presents Muirhead Inequality which is derived from the concept of majorization. It generalizes the inequality of arithmetic and geometric means. The equivalence of majorization and Muirhead¡¦s condition is illustrated. Two useful tricks for applying Muirhead Inequality are provided. Chapter 4 handles Majorization Inequality which involves Majorization and Schur convexity, two of the most productive concepts in the theory of inequalities. Its applications in elementary symmetric functions, sample variance, entropy and birthday problem are considered.

This dissertation studies several aspects of the interface between information theory and signal processing. Several new and existing results in information theory are researched from the perspective of signal processing. Similarly, some fundamental results in signal processing and statistics are studied from the information theoretic viewpoint. The first part of this dissertation focuses on illustrating the equivalence between Stein's identity and De Bruijn's identity, and providing two extensions of De Bruijn's identity. First, it is shown that Stein's identity is equivalent to De Bruijn's identity in additive noise channels with specific conditions. Second, for arbitrary but fixed input and noise distributions, and an additive noise channel model, the first derivative of the differential entropy is expressed as a function of the posterior mean, and the second derivative of the differential entropy is expressed in terms of a function of Fisher information. Several applications over a number of fields, such as statistical estimation theory, signal processing and information theory, are presented to support the usefulness of the results developed in Section 2. The second part of this dissertation focuses on three contributions. First, a connection between the result, proposed by Stoica and Babu, and the recent information theoretic results, the worst additive noise lemma and the isoperimetric inequality for entropies, is illustrated. Second, information theoretic and estimation theoretic justifications for the fact that the Gaussian assumption leads to the largest Cramer-Rao lower bound (CRLB) is presented. Third, a slight extension of this result to the more general framework of correlated observations is shown. The third part of this dissertation concentrates on deriving an alternative proof for an extremal entropy inequality (EEI), originally proposed by Liu and Viswanath. Compared with the proofs, presented by Liu and Viswanath, the proposed alternative proof is simpler, more direct, and more information-theoretic. An additional application for the extremal inequality is also provided. Moreover, this section illustrates not only the usefulness of the EEI but also a novel method to approach applications such as the capacity of the vector Gaussian broadcast channel, the lower bound of the achievable rate for distributed source coding with a single quadratic distortion constraint, and the secrecy capacity of the Gaussian wire-tap channel. Finally, a unifying variational and novel approach for proving fundamental information theoretic inequalities is proposed. Fundamental information theory results such as the maximization of differential entropy, minimization of Fisher information (Cramer-Rao inequality), worst additive noise lemma, entropy power inequality (EPI), and EEI are interpreted as functional problems and proved within the framework of calculus of variations. Several extensions and applications of the proposed results are briefly mentioned.

The intended contribution of this work is to systematically discuss a selection of methodological topics and some of the empirical and technical issues that have been driving the measurement of inequality in Mexico so far. This discussion has two strands: firstly, the general case, and second, the particular case of Mexico. The general case include some philosophical concerns, along with a review of the traditional inequality measurement, the most common operational decisions in empirical calculations, and the recent methodological contribution of development literature that is mostly centered around the capability approach of Sen (1985b). The philosophical part contrasted with other approaches and rejected the Marxist view of economic inequality, which is mostly viewed as an outcome of exploitation. The distributional judgments are compared with more ancient schools of thought in regards to justice. Another methodological issue is such that social inequality, approximated by income inequality, might be considered as an additional functioning that measures the degree of social cohesion in the country, this finding is an implication that comes from the definition of functionings within the capability approach; then, social inequality is a functioning that is different in nature from other measures of destitution, and it is also different from the destitution that is captured by absolute poverty measurement. Our general case includes a review of the most popular ways to measure inequality, such as normative and pragmatic inequality measures that are mentioned with their properties, with their rankings of the distributions provided by the use of stochastic dominance and quantile comparisons, and the construction of statistical models and some graphic representations of income economic inequality; the approach of inequality concerns included in the measurement of relative poverty is rejected for the sake of clarity. Then this general view would guide us to a better understanding of the Mexican literature for the consideration of income distribution. The measurement of destitution provided by governmental offices is necessary to discuss, because there might be some lack of coherence between the design of the measurement and the complex legal system in Mexico. We also consider a set of regulatory concerns that might not be unique to the Mexican law, but may be generalized for developing countries as a whole. Some of the methodological discussions that show how the Mexican research has been influenced by the international literature about human destitution will be good to clarify, looking at the value judgments that have been automatically accepted by the researchers. A sensitivity analysis was performed to the empirical calculation of inequality in Mexico, so the measurement showed to be different in regards to a variety of operational concerns: the recipient unit, the different data from income and consumption-expenditure surveys, various non-responses and underreported biases, the inclusion of a regional price index, among other things. In this work was also covered the reasons why it might be the case that destitution and poverty assessment was studied more deeply than inequality itself, so the possible ambiguity of inequality with poverty measurement is challenged in this work with a variety of theoretical remarks and empirical arguments. The final topic for the particular case of Mexico is to shed light in regards to the context of the capability approach and the use of equivalence scales, because these methodological approaches consider respectively directly and indirectly the assessment of distributional judgments. This discussion is followed by an empirical assessment of inequality measures that is related with a set of functionings and services, where a direct relationship of measures of inequality with other measures of destitution is made clear.

The education of youth in the United States has become a highly contested subject over the past decades. This thesis argues that one of the earliest institutions American citizens encounter – the public school system – organizes the work of many teachers in ways that reproduce inequality of opportunity for students. Drawing on qualitative data from fourteen in-depth interviews with experienced elementary school teachers in Los Angeles, this thesis illustrates how teachers experience and navigate specific structural barriers to the pursuit of equity in the classroom. Applying social reproductive theory to teacher interviews, this research discovered how, despite rhetorical commitment to equality of opportunity in education student outcomes continue to vary according to the socioeconomic status of the student population. This will help us understand systemic barriers built into the structure of the education system. These barriers operate as obstacles that teachers and students must navigate, in order to achieve success. This thesis argues that teachers should begivenmore flexibility to assess the needs of each specific class and adapt their curriculum and strategies to meet those needs. Unfortunately, in the current test-score driven system, schools with the lowest performing students are the ones whose administrations are under the most pressure to improve the low scores rather than fix the problems associated with low scores. As such, the teachers that need this flexibility the most, are the ones whose administrations keep them on the tightest rein, further reducing their ability to utilize their knowledge and implement effective strategies in the classroom. The result is the self-perpetuating cycle of inequality reproduction that we can see across North America today. / Graduate / 2019-08-13

The aim of this study was to analyze whether income inequality is a determinant of national growth and whether this influence is different when income inequality in the upper and the lower decile of income distribution are separately examined. According to the statistical analysis that was held, income inequality was found to have some statistically significant connection with the national economic growth of selected OECD countries, but only in the long run. Moreover, the research findings indicate that when a distinction is made between top-end and bottom-end income inequality, top-end inequality has a positive effect on growth, while bottom-end inequality has a negative effect. Investment and fertility rate were not found to have a statistically significant effect on growth. The above findings were evident in all four periods that were studied. The results imply that states in OECD countries, as well as countries not belonging to this group, need to pay heavy attention to bottom-end income inequality, as a means of controlling and fostering their growth potential, while at the same time leaving top-end inequalities, which not only do not undermine growth, but also drive it. Future researchers are encouraged to conduct the same research with other countries as well, especially developing ones, while also including in the research other factors moderating the effects of income inequality in growth.

In this thesis I conduct a comparative analysis of the influence of socioeconomic actors, business and labor, on public policy in Germany and the United States, specifically public policy that has an impact on economic inequality. The objective of this study is to gain a better understanding of how institutional constructs may determine the level of influence by different socioeconomic actors on public policy. In particular, I examine the link between institutional design and economic inequality, specifically the relative influence of business interests in varying types of capitalist economies and democratic systems, and assess those facets of institutional design that may facilitate the channeling of business influence in policy making. I explore institutional changes in the German political and economic system beginning in the late 1980s to determine whether these changes have altered the policy making process over time, and analyze similarities with institutional changes that have taken place in the United States beginning in the late 1970s to present. Further, I examine whether shifts in institutional design indicate that the German system is transitioning towards a more liberal model similar to that of the United States, and consider what effects this may have on the level of economic inequality in Germany. To conduct my analysis I use the Institutional Analysis and Development (IAD) framework; based on the IAD framework I create a conceptual map of the channels by which socioeconomic actors are involved in the policy making process. I evaluate the policy-making process in both formal and informal policy arenas. The policy areas analyzed include corporate governance, industrial relations, and tax, welfare and minimum wage policy during the selected time periods. The analysis shows that the institutional designs that produced the selected policies benefit business interests and may contribute towards economic inequality. The larger goal is to develop research that will build a theoretical foundation to help us identify how these systems may be improved to produce a more equitable allocation of economic resources.