  We present some product representations for random variables with the Linnik,
Mittag-Leffler and Weibull distributions and establish the relationship between
the mixing distributions in these representations. The main result is the
representation of the Linnik distribution as a normal scale mixture with the
Mittag-Leffler mixing distribution. As a corollary, we obtain the known
representation of the Linnik distribution as a scale mixture of Laplace
distributions. Another corollary of the main representation is the theorem
establishing that the distributions of random sums of independent identically
distributed random variables with finite variances converge to the Linnik
distribution under an appropriate normalization if and only if the distribution
of the random number of summands under the same normalization converges to the
Mittag-Leffler distribution.
