A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

V. Čekanavičius () and P. Vellaisamy ()
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V. Čekanavičius: Vilnius University
P. Vellaisamy: Indian Institute of Technology Bombay

Journal of Theoretical Probability, 2015, vol. 28, issue 3, 1145-1164

Abstract: Abstract We prove the Simons–Johnson theorem for sums $$S_n$$ S n of $$m$$ m -dependent random variables with exponential weights and limiting compound Poisson distribution $$\mathrm {CP}(s,\lambda )$$ CP ( s , λ ) . More precisely, we give sufficient conditions for $$\sum _{k=0}^\infty {\mathrm e}^{hk}\vert P(S_n=k)-\mathrm {CP}(s,\lambda )\{k\}\vert \rightarrow 0$$ ∑ k = 0 ∞ e h k | P ( S n = k ) - CP ( s , λ ) { k } | → 0 and provide an estimate on the rate of convergence. It is shown that the Simons–Johnson theorem holds for the weighted Wasserstein norm as well. The results are then illustrated for $$N(n;k_1,k_2)$$ N ( n ; k 1 , k 2 ) and $$k$$ k -runs statistics.

Keywords: Poisson distribution; Compound Poisson distribution; M-dependent variables; Wasserstein norm; Rate of convergence; 60F05; 60F15 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0540-5

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