On Tkachuk’s theorem for a series scheme

We consider the sum ζN = ξ1 + . . . + ξN of independent identically distributed integer-valued random variables. The problem of estimating the probability P{ζN = n} as N →∞ in the case of large deviations was solved in Tkachuk's theorem. We prove a similar theorem for the series scheme where the summand distribution depends on an unknown slowly varying function and has finite expectation and infinite variance.