Asymptotic behavior of stochastic approximation and large deviations

1983 ◽  
Author(s):  
Harold Kushner
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Stochastic Approximation

scholarly journals Precise Large Deviations for Random Sums of END Random Variables with Dominated Variation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yu Chen ◽  
Zhihui Qu
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Risk Model ◽  
Random Variables ◽  
Random Sums ◽  
Tail Probabilities ◽  
Precise Large Deviations ◽  
Surplus Process ◽  
End Random Variables ◽  
Renewal Risk

We investigate the precise large deviations for random sums of extended negatively dependent random variables with long and dominatedly varying tails. We find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence. We apply the results to a generalized dependent compound renewal risk model including premium process and claim process and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.

scholarly journals Large deviations of the limiting distribution in the Shanks–Rényi prime number race

2012 ◽  
Vol 153 (1) ◽  
pp. 147-166 ◽  
Author(s):  
YOUNESS LAMZOURI
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Linear Independence ◽  
Limiting Distribution ◽  
Absolutely Continuous ◽  
Independence Hypothesis ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where $E(x;q,a)= ({\log x}/{\sqrt{x}})(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on $\mathbb{R}^r$. Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.

scholarly journals ERGODIC PROPERTIES OF WEAK ASYMPTOTIC PSEUDOTRAJECTORIES FOR SET-VALUED DYNAMICAL SYSTEMS

2012 ◽  
Vol 13 (01) ◽  
pp. 1250011 ◽  
Author(s):  
MATHIEU FAURE ◽  
GREGORY ROTH
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Stochastic Approximation ◽  
General Class ◽  
Weak Limit ◽  
Empirical Measures ◽  
Ergodic Properties ◽  
Ergodic Behavior ◽  
Autonomous Differential Equations

A successful method to describe the asymptotic behavior of various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes is to relate them to an appropriately chosen limit semiflow. Benaïm and Schreiber (2000) define a general class of such stochastic processes, which they call weak asymptotic pseudotrajectories and study their ergodic behavior. In particular, they prove that the weak* limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic semiflow. Continuing a program started by Benaïm, Hofbauer and Sorin (2005), we generalize the ergodic results mentioned above to weak asymptotic pseudotrajectories relative to set-valued dynamical systems.

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