Often Step Stochastic Approximation Of The Long-term Status

In this thesis, the rigorous link between the long-term behavior of stochasticapproximation algorithms with constant step size and the long-term behavior of theassociated ODE are shown.In chapter2, some preliminary results in dynamical system and stochasticprocess, which will be used in this thesis, are reviewed.In the next chapter, chapter3. Under mild assumptions, it is shown that theexistence of the Feller process’s invariant measures. The criteria for the tightness ofa family of invariant measures and empirical occupations measures are proposed(cf.[20]). Along the line of research [8], if the vector field is cooperative and theequilibrium is unique, the large deviation assumptions are weaken [8], then theweak*limit of invariant measures is supported by an equilibrium p. Furthermore, themain result is obtained: the algorithms will spend most of the time in theneighborhood of a unique equilibrium p. At last, this result is applied to a class ofstochastic epidemic models (the result of [9] is extended).