Ceiling light in bedroom went on spontaneously this am. Ceiling fan in study also went on. Fan has no remote, only wall switch.

Maybe an intermittent problem, switches were already on and when power came back on, that circuit came back on?
Are they both on the same circuit?
What position were the switches in.

What kind of fan? Brand?
What kind of wall switch? Just a regular toggle?

Casablanca Inteli-Touch fans are notorious for this.