Local Limit Theorems for Large Deviations

Let (Xⱼ), j = 1, 2, , be a sequence of independent random variables with the distribution functions Vⱼ (x). We assume the existence of DXⱼ = ⱼ², sₙ² = ₉ = ₁ⁿ ⱼ², EXⱼ = 0, j = 1, 2,. We put \ Zₙ = ₉ = ₁ⁿ Xⱼ /sₙ. \ With the aid of the saddlepoint method of function theory several local limit theorems are derived, in complete analogy to the previously known integral limit theorems for large deviations of H. Cramér 1 and V. Petrov 5. These authors considered the behavior of the function P\ Zₙ 1 andx = o (n) forn. Then one has\ {pₙ䂸 (ₗ) { (x) } = e^ (x/ n) ₙ (x/ n) [ 1 + O ({x{{ n }}) }, \]where ₙ (t) is a power series converging, uniformly inn, for sufficiently small values| t |, and (x) is the density o f the normal distribution. For negative x there is a similar relation. For identically distributed Xⱼ the condition C can be considerably weakened. In this case Theorem 2 holds. Also in the case of a lattice-like distribution of the random variables Xⱼ an analogous limit relation holds (Theorem 3).