Limit theorems for transformations of sums of iid random variables (Q1382748)

Let \(X_1,X_2,\dots\) be a sequence of i.i.d. random variables with mean \(\mu\) and variance \(\sigma^2<\infty\) and define \(S_n= X_1+\cdots+X_n\). The author provides sufficient conditions on \(\varphi^{-1}(x)\) in order for the limit \[ P((\varphi(S_n)- b_1)/a_n\leq x)= \Phi(x)\tag{\(*\)} \] to hold where \(\Phi(x)\) is the normal \((0,1)\) distribution and \(a_n> 0\) and \(b_n\) are suitable constants. That is, let \(h(x)\) be a differentiable function on \((x_0,\infty)\) for some \(x_0>0\) and suppose that either \(\lim_{t\to\infty} (th'(t)/h(t))= \alpha\in(0, \infty)\) or \(\lim_{t\to\infty} (h'(t)/t^\beta h(t))= \alpha> 0\) for some \(\beta>-1\) or \(\lim_{t\to\infty} (\log t)^2/h(t)= 0\) and \(\lim_{t\to\infty} (t(\log t)h'(t)/h(t))= \alpha\in(0, \infty)\) holds, then there exist constants \(a_n> 0\) and \(b_n\) such that for \(\varphi(x)= h^{-1}(x)\) \((*)\) holds.