Sleep staging is of paramount importance in children with suspicion of pediatric obstructive sleep apnea (OSA). Complexity, cost, and intrusiveness of overnight polysomnography (PSG), the gold standard, have led to the search for alternative tests. In this sense, the photoplethysmography signal (PPG) carries useful information about the autonomous nervous activity associated to sleep stages and can be easily acquired in pediatric sleep apnea home tests with a pulse oximeter. In this study, we use the PPG signal along with convolutional neural networks (CNN), a deep-learning technique, for the automatic identification of the three main levels of sleep: wake (W), rapid eye movement (REM), and non-REM sleep. A database of 366 PPG recordings from pediatric OSA patients is involved in the study. A CNN architecture was trained using 30-s epochs from the PPG signal for three-stage sleep classification. This model showed a promising diagnostic performance in an independent test set, with 78.2% accuracy and 0.57 Cohen's kappa for W/NREM/REM classification. Furthermore, the percentage of time in wake stage obtained for each subject showed no statistically significant differences with the manually scored from PSG. These results were superior to the only state-of-the-art study focused on the analysis of the PPG signal in the automated detection of sleep stages in children suffering from OSA. This suggests that CNN can be used along with PPG recordings for sleep stages scoring in pediatric home sleep apnea tests.