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**A characterization of the weak convergence of convolution powers.**
*(English)*
Zbl 0702.60003

The classical and mostly applied method for dealing with probability measures has been the use of Fourier transforms. In several papers and two monographs [see “Weak convergence of measures.” (1982; Zbl 0538.28003), and “Limit theorems for convolutions.” (1963; Zbl 0115.103)] the reviewer instead has used Gaussian transforms and related Gaussian seminorms on the class M of finite measures \(a\mu\), \(\mu\) probability measure, a any real number, and on the class \(M'\) of signed measures generated by M. These seminorms are of interest since M is complete in the class of Gaussian seminorms and also because the Gaussian transforms are closely related to Weierstrass’ singular integral. The following theorem was proved, which is the main theorem on weak convergence of a sequence of convolution powers of probability measures to infinitely divisible probability measures:

The sequence \(\{(\mu_ n)^{*k(n)}\}\) of convolutions of probability measures is Cauchy convergent in the class of Gaussian seminorms iff \(\{k_ n(\mu_ n-e)\}\), e unit probability measure, is Cauchy convergent in the same class of seminorms.

The if-part followed by a general limit theorem which I gave for powers (and products) in a seminormed algebra. The only-if-part followed by this general theorem and necessary stability relations proved for the convergence of the convolution powers. According to the completeness, Cauchy convergence in the seminorms implies convergence in the seminorms and this convergence is equivalent to weak convergence.

Completing earlier results of F. Zitek [see Čas. Pěstováni Math. 112, 312-319 (1987; Zbl 0654.60010), ibid. 95, 62- 65 (1970; Zbl 0224.46042)] now the author considers Fourier seminorms \(F_ a(\ddot u)\) defined by \(\sup_{| t| \leq a^{-1}}| \ddot u(t)|,\) \(a>0\), for the Fourier transform \(\ddot u\) of \(\mu\) in \(M'\). He then puts \(F_ a(\mu)=F_ a(\ddot u)\) and deals with \(F_ a(\mu)\) as a seminorm on \(M'\) and proves that the main theorem for convolution powers stated above remains true if the Gaussian seminorms are changed into Fourier seminorms.

However, M is not complete in the class of seminorms \(F_ a(\mu)\) so defined. As a matter of fact, convergence of a sequence \(\{\mu_ n\}\) in these seminorms only implies convergence of the Fourier transforms and for this convergence pointwise convergence is sufficient. In order that this convergence implies weak convergence of \(\{\mu_ n\}\) it is necessary that this sequence is tight. He shows the tightness of sequences of convolution powers by the help of the necessary stability conditions and the well-known Doob’s inequality. This is his essential use of the Fourier semi-norms.

The author’s paper is of interest for comparison of the Fourier method and the direct convolution method.

The sequence \(\{(\mu_ n)^{*k(n)}\}\) of convolutions of probability measures is Cauchy convergent in the class of Gaussian seminorms iff \(\{k_ n(\mu_ n-e)\}\), e unit probability measure, is Cauchy convergent in the same class of seminorms.

The if-part followed by a general limit theorem which I gave for powers (and products) in a seminormed algebra. The only-if-part followed by this general theorem and necessary stability relations proved for the convergence of the convolution powers. According to the completeness, Cauchy convergence in the seminorms implies convergence in the seminorms and this convergence is equivalent to weak convergence.

Completing earlier results of F. Zitek [see Čas. Pěstováni Math. 112, 312-319 (1987; Zbl 0654.60010), ibid. 95, 62- 65 (1970; Zbl 0224.46042)] now the author considers Fourier seminorms \(F_ a(\ddot u)\) defined by \(\sup_{| t| \leq a^{-1}}| \ddot u(t)|,\) \(a>0\), for the Fourier transform \(\ddot u\) of \(\mu\) in \(M'\). He then puts \(F_ a(\mu)=F_ a(\ddot u)\) and deals with \(F_ a(\mu)\) as a seminorm on \(M'\) and proves that the main theorem for convolution powers stated above remains true if the Gaussian seminorms are changed into Fourier seminorms.

However, M is not complete in the class of seminorms \(F_ a(\mu)\) so defined. As a matter of fact, convergence of a sequence \(\{\mu_ n\}\) in these seminorms only implies convergence of the Fourier transforms and for this convergence pointwise convergence is sufficient. In order that this convergence implies weak convergence of \(\{\mu_ n\}\) it is necessary that this sequence is tight. He shows the tightness of sequences of convolution powers by the help of the necessary stability conditions and the well-known Doob’s inequality. This is his essential use of the Fourier semi-norms.

The author’s paper is of interest for comparison of the Fourier method and the direct convolution method.

Reviewer: H.Bergström