Geodesic
Generalized hyperbolic laws as limit distributions for random sums
Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 1, pp. 117-132

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A general theorem is proved stating necessary and sufficient conditions for the convergence of the distributions of sums of a random number of independent identically distributed random variables to one-parameter variance-mean mixtures of normal laws. As a corollary, necessary and sufficient conditions for convergence of the distributions of sums of a random number of independent identically distributed random variables to generalized hyperbolic laws are obtained. Convergence rate estimates are presented for a particular case of special continuous time random walks generated by compound doubly stochastic Poisson processes.
Keywords: random sum; generalized hyperbolic distribution; generalized inverse Gaussian distribution; mixture of probability distributions; identifiable mixtures; additively closed family; convergence rate estimate. 
@article{TVP_2013_58_1_a6,
     author = {V. Yu. Korolev},
     title = {Generalized hyperbolic laws as limit distributions for random sums},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {117--132},
     publisher = {mathdoc},
     volume = {58},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2013_58_1_a6/}
}
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V. Yu. Korolev. Generalized hyperbolic laws as limit distributions for random sums. Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 1, pp. 117-132. http://geodesic.mathdoc.fr/item/TVP_2013_58_1_a6/