Some studies have shown that neonates having experienced an Apparent Life-Threatening Event (ALTE) have an increased risk of dying of the Sudden Infant Death Syndrome (SIDS). In this study the heart rate variability in neonates has been analyzed through a study of the long range correlations and scale invariance (self-similarity) in their cardiac rhythm. The goal is to see whether it is possible to distinguish normal infants from infants with ALTEs using the previously mentioned nonlinear measures of the heart rate variability. Twelve ALTE infants are included in this study. To determine the degree to which these neonates suffer from ALTEs two examinations are performed: a polysomnography and home monitoring. Using the nonlinear measures obtained from the nonlinear analysis of the heart rate variability, the group suffering most from ALTEs can clearly be separated from the other neonates.