Consider a double array \(\left\{ {X_{i,j} ;i \geqslant 1,j \geqslant } \right\}\) of i.i.d. random variables with mean μ and variance \(\sigma ^2 (0 < \sigma ^2 < \infty )\) and set \(Z_{i,n} = n^{ - 1/2} \sum\nolimits_{j = 1}^n {(X_{i,j} - \mu )} /\sigma \). Let \(\hat \Phi _{N,n} \) denote the empirical distribution function of Z1, n,..., ZN, n and let Φ be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for \(\sqrt N (\hat \Phi _{N,n} - \Phi )\), where n=n(N)→∞ as N→∞. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.