Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random , provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples

Sharp convergence for degenerate Langevin dynamics

In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is sharp, that is, the total variation distance of the accelerated random dynamics and its equilibrium distribution tends to a decreasing profile, which corresponds to the total variation distance between the marginal of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. In particular, there is no cutoff phenomenon for this one-parameter family of random processes.Comment: 37 page

Superdiffusive planar random walks with polynomial space-time drifts

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent 3 / 4 . The self-interacting process originated in discussions with Francis Comets

Dos Oceanos para as telas: investigando o estereótipo do tubarão

TCC(graduação) - Universidade Federal de Santa Catarina. Centro de Ciências Biológicas. Biologia.Inspirado por uma aula que viveu na graduação, o autor investiga o tema dos estereótipos ao longo deste trabalho, principalmente aqueles ligados aos tubarões. Para refletir e investigar sobre isso, ele aciona pensamentos sobre cultura, artefatos midiáticos, cinema, Estudos Culturais, entre outros, e suas implicações em nossas subjetividades e na sociedade. Fez-se necessário refletir também sobre como nós, seres humanos, nos relacionamos com os outros seres do Reino Animal. Quais são os impactos de colocarmos os nossos adjetivos - humanos - nos animais não humanos? O trabalho culmina em análises de cartazes e sinopses de filmes hollywoodianos sobre os tubarões. É a partir de suas sensações e impressões que o autor tece suas análises acionando os estudos feitos ao longo da sua trajetória, pensando sempre na questão do estereótipo e na influência que essas imagens possuem em nosso imaginário como sociedade. Por fim, ele nos apresenta reflexões sobre as relações entre humanos e tubarões, e pensamentos que nos instigam a brincar com os estereótipos (como por exemplo as animações), quebrá-los, rasurá-los, para quem sabe, permitir outros modos de nos relacionar com os animais não humanos e com o meio ambiente