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Stable convergence and applications. (Convergence stable et applications.) (French) Zbl 0953.60003
Stable convergence goes back to A. Rényi, cf. his book “Probability theory” (1970; Zbl 0206.18002). He showed that the central limit theorem also holds for conditional distributions given any event $$B$$ with $$P(B)>0$$. Following J. Jacod and J. Mémin [in: Séminaire de probabilités XV, Lect. Notes Math. 850, 529-546 (1981; Zbl 0458.60016)] and E. J. Balder [J. Math. Anal. Appl. 72, 391-398 (1979; Zbl 0434.49007) and Math. Oper. Res. 13, No. 2, 265-276 (1988; Zbl 0658.90104)] the present survey paper puts stable convergence into a general setting. Let $$(\Omega,{\mathcal A},P)$$ with $$P$$ fixed be a probability space and $$E$$ with Borel $$\sigma$$-algebra $${\mathcal E}$$ a Polish space. A Markov kernel $$N(\omega,A)$$, $$\omega\in\Omega$$, $$A \in {\mathcal E}$$, defines probability measures $$QN$$ with $$QN(A)= \int_\Omega Q(d \omega)N (\omega,A)$$ on $${\mathcal E}$$ and $$P\times N$$ with $$\int fdP \times N= \int_\Omega P(d\omega) \int_EN(\omega, dx)f (\omega,x)$$ on $${\mathcal A} \times {\mathcal E}$$. Then $$P\times N$$ is a general Young measure on $${\mathcal A} \times {\mathcal E}$$ with projection $$P$$ on $${\mathcal A}$$. The $$P$$-weak topology and convergence for kernels then is defined. The $${\mathcal U}$$-stable convergence of a family $$X(t)$$ of $$E$$-valued random variables on $$\Omega$$ is the weak convergence of the kernels $$N(t)$$ to $$N$$. Here $$N(t)$$ is a regular version of the conditional distribution of $$X(t)$$ given $${\mathcal U}\subset {\mathcal A}$$. When $$N$$ is the conditional distribution of $$X$$ given $${\mathcal U}$$ we have $${\mathcal U}$$-stable convergence of $$X(t)$$ to $$X$$. This is intermediate between convergence in law and in probability. Conditions for weak and stable convergence and relative weak compactness (tightness) are proved. An application to convergence of a normal law when $$E=R$$ is given.
Reviewer: A.J.Stam (Winsum)

##### MSC:
 60F05 Central limit and other weak theorems 60B05 Probability measures on topological spaces 60B10 Convergence of probability measures