## 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