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On asymptotics of distribution of the ratio of sums of random variables. (Russian) Zbl 0719.62027
This paper deals with the limit distribution of the ratio \(\xi_ 1+...+\xi_ n/\eta_ 1+...+\eta_ n\) where the couple \((\xi_ j,\eta_ j)\) consists of i.i.d. random variables. It also contains some applications of the obtained results to the asymptotic behaviour of \[ \int_{R^ n}[f(x_ 1)+...+f(x_ n)]/[g(x_ 1)+...+g(x_ n)]\mu (dx_ 1,...,dx_ n) \] where \(\mu\) is a finite measure and f and g are \(\mu\)-integrable functions. Moreover the authors investigate the problem of statistical estimation of the parameter \(\alpha\) of the distribution \[ F_{\alpha}(x)=1-\ell (x)x^{-\alpha}I(x>0), \] where \(\ell (x)\) is a slowly varying function.

62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
60F05 Central limit and other weak theorems