On the stability problem for conditional expectation
نویسندگان
چکیده
The behavior of the conditional expectation E{X I Y) under a small perturbation Z of the conditioning random variable Y is analyzed. We show that if Y and Z are independent then E(XIY + EZ) converges to E{XIY) in mean as E --* 0 for all integrable X, provided the distribution of Y is absolutely continuous. We also show that the limit is E{X 1 Y, Z} rather than E(X 1 Y}, i.e., there is no stability, when Y is a discrete (i.e., countably valued) random variable. Finally, we show that in general E(X ( Y + EZ) might have no limit in distribution as E * 0. AMS 1980 Subject Classification: 6OH99. Kqvwords: Conditional expectations; stability; perturbation.
منابع مشابه
Efficient Simulation of a Random Knockout Tournament
We consider the problem of using simulation to efficiently estimate the win probabilities for participants in a general random knockout tournament. Both of our proposed estimators, one based on the notion of “observed survivals” and the other based on conditional expectation and post-stratification, are highly effective in terms of variance reduction when compared to the raw simulation estimato...
متن کاملCONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS
The notion of a $D$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of random variables on theK^{o}pka's $D$-Poset and prove the basic properties ofconditional expectation on this structure.
متن کاملOn Psi-conditional asymptotic stability of first order nonlinear matrix Lyapunov system
We provide necessary and sucient conditions for psi-conditional as-ymptotic stability of the solution of a linear matrix Lyapunov system and sucientconditions for psi -conditional asymptotic stability of the solution of a rst ordernon-linear matrix Lyapunov system X0 = A(t)X + XB(t) + F(t;X).
متن کاملOn reducibility of weighted composition operators
In this paper, we study two types of the reducing subspaces for the weighted composition operator $W: frightarrow ucdot fcirc varphi$ on $L^2(Sigma)$. A necessary and sufficient condition is given for $W$ to possess the reducing subspaces of the form $L^2(Sigma_B)$ where $Bin Sigma_{sigma(u)}$. Moreover, we pose some necessary and some sufficient conditions under which the subspaces of the form...
متن کامل