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Doctoral Thesis
Full name
Vitor Gustavo de Amorim
Knowledge Area
Date of Defense
São Carlos, 2022
Gallo, Alexsandro Giacomo Grimbert (President)
Chazottes, Jean-René
Freitas, Ana Cristina Moreira
Galves, Jefferson Antonio
Vares, Maria Eulalia
Title in English
Poincaré recurrence times in stochastic mixing processes
Keywords in English
Exponential approximation
Hitting and return times
Mixing processes
Potential well
Waiting time spectrum
Abstract in English
In the context of the discrete-time stochastic processes, this thesis presents new results on Poincaré recurrence theory. After a complete review of recent results, we present a new theorem on the exponential approximations for hitting and return times distributions. We show that the scaling parameter of the approximate distribution, called potential well, brings fundamental informations about the structure of the target set. Moreover, we show that the asymptotic properties of the potential well influences several aspects of the recurrence times, such as limiting distributions and moments. Finally, we apply our results to obtain the waiting time spectrum as a function of the Rényi entropy for classes of processes not covered by previous works.
Title in Portuguese
Tempos de recorrência de Poincaré em processos estocásticos com mistura
Keywords in Portuguese
Aproximação exponencial
Espectro do tempo de espera
Poço de potencial
Processos misturadores
Tempos de entrada e retorno
Abstract in Portuguese
Esta tese apresenta novos resultados na teoria de recorrência de Poincaré, no contexto de processos estocásticos em tempo discreto. Após uma revisão completa de resultados recentes, apresentamos um novo teorema sobre aproximação exponencial para distribuições de tempos de entrada e retorno. Mostramos que o parâmetro de escala da distribuição aproximada, chamado poço de potencial, traz ao teorema de aproximação informações fundamentais sobre a estrutura do conjunto alvo. Além disso, mostramos que as propriedades assintóticas do poço de potencial influenciam vários aspectos dos tempos de recorrência, como as distribuições limite e seus momentos. Por fim, aplicamos nossos resultados para obter o espectro do tempo de espera como função da entropia de Rényi para classes de processos nã cobertas por trabalhos anteriores.
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