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An invariance principle for processes with conditionally independent increments. (English) Zbl 1029.60509

Àrato, M. (ed.) et al., New trends in probability and mathematical statistics. Proceedings of the second Ukrainian-Hungarian conference, Mukachevo, Ukraine, September 25-October 1, 1992. Kiev: TViMS. Teor. Veroyatn. Mat. Stat./Probab. Theory Math. Stat. 2, 404-409 (1995).
Let for every \(n \geq 1\) \(\{ \eta_n(k)\}_{k \geq 1}\) be a sequence of random variables with values in a certain measurable space \(\{ X,{\mathcal B}_X\}\) and let \(\{ \xi_n(k,x), x\in X\}_{k \geq 1}\) be a family of independent random variables. These sequences are independent. Limit behaviour of the random processes \[ \zeta_n(t)= B_n^{-1} \sum_{t_k <t} \xi_n(k,\eta_k(t)), \] where \[ B_n^2 =\sum_{k=1}^n E\sigma_n^2(k,\eta_k(t)), \qquad \sigma_n^2(k,x) = E\xi_n^2(k,x) <\infty, \]
\[ t_i =t_{ni}=B_n^{-2}(z)\sum_{k=1}^i E\sigma_n^2(k,\eta_n(k)). \] is investigated in the paper. Let \(F_{nj}^i =\sigma\{ \eta_n(k)\), \(j\leq k \leq i\}\) be a \(\sigma\)-algebra generated by random variables \(\eta_n(k)\), \(j\leq k \leq i.\) Let \(\varphi(r) = \sup\{ |P(B/A) -P(B) |: A \in F_{n,1}^m\), \(B\in F_{n,m+r}^\infty\), \(n,m \geq 1\}\) be the coefficient of the uniform strong mixing condition for the random sequence \(\eta_n(k).\) Under some assumptions on \(\varphi(r)\) the authors find conditions under which \(\xi_n(t)\) converges weakly in the space with Skorokhod \(J_1\)-topology to the standard Wiener process \(w(t)\) as \(n \to \infty.\) The case when the random variables \(\xi_n(k,x)\) are dependent is considered too.
For the entire collection see [Zbl 0896.00028].

MSC:

60F17 Functional limit theorems; invariance principles
60J65 Brownian motion
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