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Positive martingales and random measures. (English) Zbl 0636.60049

Let \((Q_ n(t))\), \(n=0,1,...\), be a positive martingale indexed by t (t\(\in T\) compact metric space) and let \(\sigma\) be a measure on T (\(\sigma\in M\) \(+(T))\). \((Q_ n\sigma)\), \(n=0,1,...\), is a sequence of random measures. If \(\int_{T}E Q_ n(t)d\sigma (t)<\infty\) the random measures \(Q_ n\sigma\) converge weakly a.s. to a random measure Q \(\sigma\). Conditions are given to insure either E Q \(\sigma\) \(=0\) or E Q \(\sigma\) \(=\sigma\) (in general E Q \(\sigma\leq \sigma)\). Examples and applications are given (random coverings, Mandelbrot martingales, multiplicative chaos).
Reviewer: L.Gal’čuk

MSC:

60G44 Martingales with continuous parameter
60G57 Random measures
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