Just to "pick a nit", what every one is calling an "analog" TV should be called a TV with NTSC (National Television System Committee) tuner. Newer TVs have a ATSC (Advanced Television Systems Committee) tuner you receive. Newer TV also have a second tuner, QAM (Quadrature amplitude modulation) to receive digital cable channels.
Yes, that is right. You do NOT need a "set top box" DVR to receive a digital signal from your cable company. They tell you that one is required, but it is only required because the "scramble" the signal they put on the wire. You can prove that by hooking your TV directly up to the cable and rescan for channels. You will be able to view all of the local broadcast channels that are required to be sent over the wire UNSCRAMBLED !