An estimation of the stability and the localisability functions of multistable processes

Multistable processes are tangent at each point to a stable process, but where the index of stability and the index of localisability varies along the path. In this work, we give two estimators of the stability and the localisability functions, and we prove the consistency of those two estimators. We illustrate these convergences with two classical examples, the Levy multistable process and the Linear Multifractional Multistable Motion

The Hausdorff dimension of the range of the Lévy multistable processes

International audienceWe compute the Hausdorff dimension of the image X(E) of a non random Borel set E ⊂ [0, 1], where X is a Lévy multistable process in R. This extends the case where X is a classical stable Lévy process by letting the stability exponent α be a smooth function, which leads to non-homogeneous processes because their increments are not stationary and not necessarily independent. Contrary to the situation where the stability parameter is a constant, the dimension depends on the version of the multistable Lévy motion when the process has an infinite first moment

Localisable moving average stable and multistable processes

We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a ``tangent process'', in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein-Uhlenbeck process and the multistable reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is, at each time t, a L\'evy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations

Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes

International audienc

Exponential inequalities for the supremum of some counting processes and their square martingales

We establish exponential inequalities for the supremum of martingales and square martingales obtained from counting processes, as well as for the oscillation modulus of these processes. Our inequalities, that play a decisive role in the control of errors in statistical procedures, apply to general non-explosive counting processes including Poisson, Hawkes and Cox models. Some applications for $U$-statistics are discussed

Odontoblasto : uma célula sensorial ao serviço na nocicepção

Dissertação para obtenção do grau de Mestre no Instituto Universitário Egas MonizO odontoblasto diferencia-se durante a odontogénese e desempenha diferentes funções durante toda a vida do órgão dental. Contribuem primeiro para a formação e mineralização do dente, garantindo a proteção do dente pela produção de dentina reacional. Estas células ciliadas também parecem desempenhar um papel na transmissão da dor. A sua localização espacial única, em estreita relação com as terminações nervosas da polpa, sugere que possam desempenhar um papel fundamental na transdução sensorial dos eventos que ocorrem no interior da dentina. Esta revisão narrativa pretende estudar a sensorialidade potencial dos odontoblastos, analisando a expressão de receptores sensoriais e mecanismos moleculares subjacentes, permitindo a percepção e transmissão de sinais nociceptivos da dor dentinária. Ao nível dos métodos, para esta revisão narrativa, foi feita uma pesquisa bibliográfica, utilizando os motores de pesquisa Pubmed/MEDLINE, Science Direct, Scopus, Cochrane Library e outras fontes secundárias. Utilizámos artigos em português, espanhol, inglês e francês disponíveis em texto completo publicados nos últimos 15 anos e ainda outros estudos baseados em evidências.The odontoblast differentiates itself during dentistry and performs different functions throughout the life of the dental organ. They first contribute to the formation and mineralization of the tooth, ensuring the protection of the tooth by the production of reactional dentin. These ciliated cells also seem to play a role in the transmission of pain. Their unique spatial location, in close relationship with the nerve endings of the pulp, suggests that they can play a key role in the sensory transduction of events occurring inside the dentin. This narrative review aims to study the potential sensoriality of odontoblasts by analysing the expression of sensory receptors and underlying molecular mechanisms, allowing the perception and transmission of nociceptive signals of dentin pain. In terms of methods, for this narrative review, a bibliographic search was made using Pubmed/MEDLINE, Science Direct, Scopus, Cochrane Library and other secondary sources. We have used articles in Portuguese, Spanish, English and French available in full text published in the last 15 years as well as other evidence-based studies