Rate of convergence of random sums of random vectors to Marshall-Olkin distribution
Let \((\xi_ i,\eta_ i)\), \(i=1,2,...\), be independent nonnegative identically distributed random vectors, and let \((\nu_ 1,\nu_ 2)\) be random vectors independent of \((\xi_ i,\eta_ i)\), \(i=1,2,...\), having the two-dimensional geometrical distribution \(q^{\ell -1}(p_ 2-p_{12})(1-p_ 1)^{k-\ell -1}p_ 1,\) \(k>\ell,\) \(P\{\nu_ 1=k,\nu_ 2=\ell \}=\{\) \(q^{k-1}p_{12},\) \(k=\ell\), \(q^{k-1}(p_ 1-p_{12})(1-p_ 2)^{\ell -k-1}p_ 2,\) \(k<\ell,\) where k and \(\ell\) are positive integers, \(p_ 1\), \(p_ 2\) and \(p_{12}\leq \min (p_ 1,p_ 2)\) are probabilities, and \(q=1-p_ 1- p_ 2+p_{12}>0\). Let \[ G(x,y)=P\{p_ 1\sum^{\nu_ 1}_{i=1}\xi_ i<x,\quad p_ 2\sum^{\xi_ 2}_{i=1}\eta_ i<y\}, \] \[ \lambda_ 1=\lim_{q\to 1}(p_{12}/p_ 1),\quad \lambda_ 2=\lim_{q\to 1}(p_{12}/p_ 2) \] and \[ \mu_ 1^{(2)}=E \xi^ 2_ 1,\quad \mu_ 2^{(2)}=E \xi^ 2_ 2. \] Then the authors prove the following result related to rate of convergence. Theorem. If E \(\xi\) \({}_ 1=E \xi_ 2=1\) and \(\mu_ 1^{(2)}<\infty\), \(\mu_ 2^{(2)}<\infty\), then there exists an absolute constant C such that \[ \sup_{x,y\geq 0}\inf_{\lambda_ 1,\lambda_ 2\geq 0}| G(x,y)-1+e^{-x}+e^{-y}-e^{-(1-\lambda_ 1)x-(1-\lambda_ 2)y-\max (\lambda_ 1x,\lambda_ 2y)}| \leq \] \[ \leq C\{p_ 1\mu_ 2^{(2)}+p_ 2\mu_ 2^{(2)}+p_{12}(\mu_ 1^{(2)} \ln^ 2 p_ 1+\mu_ 2^{(2)} \ln^ 2 p_ 2)\}. \]
scientific article; zbMATH DE number 4215047