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We prove large deviation results for the random sum 
, 
, where 
are non-negative integer-valued random variables and are 
i.i.d. non-negative random variables with common distribution function F, independent of 
. Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.
