How can you tell whether there's a signal (when your signal looks a lot like noise)?
So far you seemed to identify snoring through the presence of a strong periodic pattern in the audio. If you tell me that there might be other sources with the same property, it's time to move on and focus on a property of the signal more specific of snoring ; and I would suggest to look more precisely into the timbre of the recorded sound itself. Given that a human listener can tall apart snoring from breathing, you might resort to traditional audio signal classification approach. Compute features like MFCCs and Fluctuation patterns on your audio data, and train a bayesian classifier (or more sophisticated) to discriminate snoring from breathing.

It might be time also to review the literature on the topic... For example I found this: http://web.itu.edu.tr/kamasak/pubs/pdf/pm7_8_007.pdf