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[For the entire collection see Zbl 0718.00011.]
Lepage observed that the well-known series expansion of the Wiener integral \(\int fdW\), where \(W\) is a Wiener process, and \(f\) is a square integrable function, is, in a sense, valid also in the case of a symmetric \(\alpha\)-stable integrator. If \(M\) is a Lévy stable motion, \(0<\alpha < 2\), \(f\) is \(\alpha\)-integrable, then \(\int^ 1_ 0 fdM\), and \(const \sum_ n\varepsilon_ n f(U_ n)\Gamma^{-1}\) are equidistributed, where \((\varepsilon_ n)\) is a Rademacher sequence, \(\Gamma_ n\) are arrival times of a standard Poisson process, and \((U_ n)\) is a sequence of i.i.d. uniform random variables on [0,1], and three sequences are mutually independent. The Lepage’s representation can be generalized to symmetric stable independently scattered integrators on abstract measure spaces via a measure theoretical isomorphism, which slightly complicates the appearance of the random series. A suitable multiple series representation of this kind was obtained by the first author and the reviewer for multiple symmetric stable integral. Series representations are valid for general pure jump Lévy processes.
The article reverses, so to speak, the aforementioned constructions. First, the Lepage-type series for indicators of sets in the Cartesian power of an abstract measure space are shown to induce a product random (vector) measure. The proof is quite cumbersome. Then, the construction of the integral proceeds as usual, via approximation by simple functions, for which the integral is easily defined.