I feed the TV with HDMI through the wall. I don't know what type of "digital converter" you're referring to, but only STBs from Verizon will work to get Verizon programming. "Digital converters" for over-the-air will not work unless they have a QAM tuner. Some TVs have QAM receivers and will receive channels 1-49 and local HD channels from Verizon using a direct coax feed with no other box.
I get Verizon programming from my TiVos. No need for a Verizon FiOS STB.
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