Asymptotic properties of multidimensional stable densities and asymmetric problems of large deviations

The paper considers asymptotic properties of the so-called asymmetric multidimensional stable distributions with the property that the minimal convex conus generated by a support of Poisson spectral measure does not coincide with $\mathbf R^d$. The density of such a distribution along some directions can decrease extremely quickly. Using methods of the conjugate Cramér distributions we find the exact asymptotic and write an asymptotic series which describes a character of the decrease.