Stable convergence and applications. (Convergence stable et applications.) (French) Zbl 0953.60003

Letta, G.

Atti Semin. Mat. Fis. Univ. Modena 46, Suppl., 191-211 (1998).

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

Keywords:

weak convergence; stable convergence; Young measure; Markov kernel; conditional distribution