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Uniqueness theorems for Gaussian measures in \(\ell _ q\), \(1\leq q<\infty\). (English) Zbl 0618.60005

We prove that a Gaussian measure \(\mu\) in \(\ell _ q\) is uniquely determined by the function \[ y\to \int _{\ell _ q}\| x+y\| ^ p_ qd\mu (x),\quad y\in \ell _ q, \] iff \(p\neq q\). This follows from properties of the mapping \[ s\to \int _{\ell _ q}\| x+sy\| ^ p_ qd\mu (x),\quad s\in {\mathbb{R}}, \] where y is either a fixed unit vector of \(\ell _ q\) or the sum (or difference) of two such vectors. Besides \(q=1\) or \(q=2\) this function describes the distribution of the coordinate functional generated by this unit vector.

MSC:

60B05 Probability measures on topological spaces
60G15 Gaussian processes
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References:

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