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The integral limit theorem as to the probability distribution of the random number νm of summands in the sum ∑k=1νmξk is proved. Here, ξ1,ξ2,… are some nonnegative, mutually independent, lattice randomvariables being equally distributed and νm is defined by the condition that the sum value exceeds at the first time the given level m∈ℕ when the number of terms is equal to νm

The integral limit theorem in the first passage problem for sums of independent nonnegative lattice variables