Abstract
This chapter discusses weak convergence in separable Hilbert spaces. If a linear space has a scalar product and a norm and is complete in the metric, it is called a Hilbert space. Any separable Hilbert space is isometric to l2; hence, the l2-space is a representative of a rather general class of Hilbert spaces. A weak limit problem for sequences of measures on a separable Hilbert space can then be transformed to a corresponding limit problem on l2. The chapter presents weak convergence of convolution products of probability measures on l2. It describes necessary and sufficient conditions for the weak convergence of convolution products of symmetrical probability measures.
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