Something is wrong here. I may get a few "dead air" type of phone calls, but most of them have a recorded robotic voice that appears when I say something or when I pickup the line. Telemarketers might be evil, but they're not stupid. They would not waste the cost of a call just to deliver "dead air". Certainly not for 4 years of "dead air". Something is wrong.
My guess(tm) is something is wrong with your Verizon POTS line that is initiating a ring, but not completing the call. I've seen this with some electronic phones, where there is sufficient crosstalk in the wire bundle to pickup some of the ringing voltage from other lines in the bundle. However, those don't also pass Caller ID numbers and only ring a few phones in the house, so that's not a likely failure mode. Unless the provisioning is mangled or the Verizon switch has gone insane, I can't guess(tm) what might be causing the calls.
I was thinking it might be a fax machine trying to send a fax repeatedly, but that would be from one phone number and certainly not for 4 years. You would also hear a tone as the originating fax machine tries to negotiate the call. Are the numbers shown on the Caller ID all identical or perhaps similar as from a calling group?
I assume that you've contacted Verizon. Changing your phone number might be an obvious option that I'm sure they would have suggested. If the problem persists, it's a hardware or switch problem. If it goes away, problem solved.
On the other hand, the vague problem description, improbable symptoms, and odd selection of crossposted newsgroups leads me to suspect that this is some manner bad joke or time burner. Please assure me this problem is real by posting some details.