Convergence of Laws and Central Limit Theorems

Let X_j be independent, identically distributed random variables with mean 0 and variance 1, and S_n := X ₁ + ⋯ + X_n as always. Then one of the main theorems of probability theory, the central limit theorem, to be proved in this chapter, states that the laws of S_n/n^1/2 converge, in the sense that for every real x,