Brownish black spots on the leaves or around the edges of the leaves can be a water stress issue.
Too much or too little can result in these black spots. Your soil should be kept moist to the touch but not soggy. If you live in a place with dry air you may need to humidify the air.

Over fertilization could also be a problem. Use a slow release granular fertilizer once a month or use a water soluble fertilizer every two weeks. It is best to reduce watering and fertilizer during the winter months.

The only other thing that could cause this problem is fungal leaf spot. Over watering can cause this problem as well. Remove leaves that have the black spots and avoid misting the plant directly. Watering the leaves can spread the fungus. If the fungus continues to spread after removing the infected leaves, you will need to get a fungicide. Your local garden center nursery will have what you need.