  We present new mixture representations for the generalized Linnik
distribution in terms of normal, Laplace, exponential and stable laws and
establish the relationship between the mixing distributions in these
representations. Based on these representations, we prove some limit theorems
for a wide class of rather simple statistics constructed from samples with
random sized including, e. g., random sums of independent random variables with
finite variances and maximum random sums, in which the generalized Linnik
distribution plays the role of the limit law. Thus we demonstrate that the
scheme of geometric (or, in general, negative binomial) summation is far not
the only asymptotic setting (even for sums of independent random variables) in
which the generalized Linnik law appears as the limit distribution.
