On the speed of convergence in the local limit theorem for lattice distributions

The paper deals with the speed of convergence in the local limit theorem for lattice distributions. Numerical calculations have been carried out for the example of random variables taking on values $-3$, 0, 7 with probability $1/3$ each. It follows as a result of these calculations that the behaviour of the probabilities $P_n(k)$ is much less regular than one might have expected. Their “smoothing” which should take place according to the local limit theorem occurs when $n$ is very large.
An estimate of the number of summands necessary to achieve the prescribed accuracy of the normal approximation to $P_n(k)$ is also given.