  An improved version of the functional limit theorem is proved establishing
weak convergence of random walks generated by compound doubly stochastic
Poisson processes (compound Cox processes) to L{\'e}vy processes in the
Skorokhod space under more realistic moment conditions. As corollaries,
theorems are proved on convergence of random walks with jumps having finite
variances to L{\'e}vy processes with variance-mean mixed normal distributions,
in particular, to stable L{\'e}vy processes, generalized hyperbolic and
generalized variance-gamma L{\'e}vy processes.
