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Functional limit theorems in the \(\mathcal M\)-scheme. (English. Russian original) Zbl 0892.60046

Lith. Math. J. 37, No. 2, 108-118 (1997); translation from Liet. Mat. Rink. 37, No. 2, 139-154 (1997).
By analogy with the theory of sums of independent random variables (the authors call \({\mathcal A}\)-scheme), one-dimensional limit distributions of the arrays of products of independent infinitesimal r.v.s (the authors call \({\mathcal M}\)-scheme) were studied by Abramov (1986, 1987), Bakštis (1968, 1971 and 1972) and Zolotarev (1957, 1962). Having in mind the well-known invariance principles in \({\mathcal A}\)-scheme, the present authors start to deal with functional limit theorems for step functions defined via products of conditional infinitesimal independent r.v.s. Using tightness and the concept of measures on functional spaces as presented in the books of Billingsley (1966) and Skorokhod (1964), the authors are able to prove the convergence of the sequence of stochastic processes in \({\mathcal M}\)-scheme to a stochastically continuous process with independent quotients.

MSC:

60F17 Functional limit theorems; invariance principles
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References:

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