Abstract:
The paper provides necessary and sufficient conditions for weak convergence of the distributions of sums of independent random variables to normal and Poisson distributions. The conditions are given in terms of the first through fourth moments of the truncated components and some condition guaranteeing the asymptotic equality between the distribution of the sums mentioned above and the distribution of the sums of the truncated components. The theorems proven in this work are similar to the well-known criteria for normal and Poisson convergencies.
Keywords:normal convergence, Poisson convergence, Lindeberg–Feller central limit theorem, weak convergence, moments of random variables.
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