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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 27, Fasc. 2,
pages 153 - 166
 

FINITE DIFFERENCE EQUATIONS AND CONVERGENCE RATES IN THE CENTRAL LIMIT THEOREM

Lars Lindhagen

Abstract: We apply the theory of finite difference equations to the central limit theorem, using interpolation of Banach spaces and Fourier multipliers. Let $S_n^*$ be a normalized sum of i.i.d. random vectors, converging weakly to a standard normal vector $\mathcal (N)$. When does $\| Eg(x+S_n^*)- Eg(x+\mathcal (N))\|_(L_p(dx))$ tend to zero at a specified rate? We show that, under moment conditions, membership of $g$ in various Besov spaces is often sufficient and sometimes necessary. The results extend to signed probability.

2000 AMS Mathematics Subject Classification: Primary: 46B70, 60F05, 65M06; Secondary: 35K05, 42B15, 65M15;

Key words and phrases: Finite difference equations, central limit theorem, convergence rate, interpolation theory, Fourier multipliers, Besov spaces, signed probability.

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