Limit Theorems for Large Deviations

On the necessity of Statulevičius' condition in limit theorems for large-deviation probabilities

“…Proof. Let us consider the first of relations (2). For p(n) ~< x ~< A2(1 --exp{ ,t/,+_~• p,(n) }) we have from (3) and (4) with en --+ 0 and e', ~ 0, whereas…”

“…where y = O, in the region (2) in the region 0 ~< x < AVz 1/(2(1+2y)) and necessary when A is replaced A0 in (1), where Ao is defined in Theorem 1. Proof.…”

Large deviations for integer centered poisson approximation

“…For the normal law the Statulevi~ius condition (S) allows one to use a unified treatment for the sums of independent and dependent summands. Such an approach has proved to be very fruitful (see, for example, [23]). In this section, we shall use one analogue of the condition (S) for the lattice variables from the paper by Ale~kevi~ien6 and Statulevieius [3].…”

Poisson approximation for the number of large digits of inhomogeneousf-expansions

“…The crucial point in applying the technique developed in [7] Note that the symbol E was introduced by V. Statulevi~ius in the Sixties in a different way in order to represent the mixed cumulants of dependent random variables, see SAULIS and STAXtrcEVI~mS [17].…”