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oct 2000 Z-theorems: limits of stochastic equations
Vladimir V. Anisimov, Georg CH. Pflug
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Bernoulli 6(5): 917-938 (oct 2000).


Let fn(θ,ω) be a sequence of stochastic processes which converge weakly to a limit process f0(θ,ω). We show under some assumptions the weak inclusion of the solution sets θ n (ω)={θ:f n(θ,ω)=0} in the limiting solution set θ 0 (ω)={θ:f 0(θ,ω)=0} . If the limiting solutions are almost surely singletons, then weak convergence holds. Results of this type are called Z-theorems (zero-theorems). Moreover, we give various more specific convergence results, which have applications for stochastic equations, statistical estimation and stochastic optimization.


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Vladimir V. Anisimov. Georg CH. Pflug. "Z-theorems: limits of stochastic equations." Bernoulli 6 (5) 917 - 938, oct 2000.


Published: oct 2000
First available in Project Euclid: 6 April 2004

zbMATH: 0965.60036
MathSciNet: MR2002B:60028

Keywords: asymptotic distribution , consistency , stochastic equations , stochastic inclusion

Rights: Copyright © 2000 Bernoulli Society for Mathematical Statistics and Probability


Vol.6 • No. 5 • oct 2000
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