Existence and Spectrality of random measures generated by infinite convolutions

In this paper, we construct a class of random measures ^n by infinite convolutions. Given infinitely many admissible pairs \ (N₊, B₊) \₊=₁^ and a positive integral sequence n=\n₊\₊=₁^, for every N^N, we write ^n () = ₍__{₁^-n₁B_₁} * ₍__{₁^-n₁N_₂^-n₂B_₂} *. If n₊=1 for k 1, write () =^n (). First, we show that the mapping ^n: (, B) ^n () (B) is a random measure if the family of Borel probability measures \ (): N^{N\} is tight. Then, for every Bernoulli measure P on N^N, the random measure ^n is also a spectral measure P-a. e. . If the positive integral sequence n is unbounded, the random measure ^n is a spectral measure regardless of the measures on the sequence space N^N. Moreover, we provide some sufficient conditions for the existence of the random measure ^n. Finally, we verify that random measures have the intermediate-value property.