Local theorems for the maximum of sums of independent identically distributed random variables

B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables [in Russian], Moscow-Leningrad (1949). S. V. Nagaev, “On convergence rate in a certain boundary value problem, I,” Teoriya Veroyat. i ee Primen.,15, No. 2, 179–199 (1970). S. V. Nagaev, “Some limit theorems for large deviations,” Teoriya Veroyat. i ee Primen.,10, No. 2, 231–354 (1965). A. A. Mitalauskas and V. A. Statulyavichus, “Local limit theorems and asymptotic expansions for sums of independent lattice random variables,” Liet Matem. Rink.,6, No. 4 (1966). E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford (1937).