Hi, I think I know why this could be happening. The remotes typically have a code to minimize interference with other remote controlled devices like garage door openers. Does your neighbor have a remote controlled light perhaps? Two of the fans we installed do, so it isn't completely strange.
You could change the code on your remote and receiver and see what happens. On our fans the receiver was inside the canopy at the top of the fan and there is a little panel of 4 tiny switches. You'll put some of them up and some of them down and match the remote to the same pattern. The panel on our remote was under the battery cover.
This would be a relatively simple thing and wouldn't require an electrician--you wouldn't be messing with wiring at all. If it doesn't fix the problem, I would call an electrician at that point.
Re: Mira re: your fan turning on neighbor's lights
In related news: Our neighbors garage door opener started opening our garage door. It stopped after the H reprogrammed our openers.