The mortgage requirements will vary depending if you are purchasing the property as a primary residence or if you are buying it as an investment property. John Lajara is correct, if you plan on using the property as a primary residence you will have the option of putting 3.5% down or 20%. If you have excellent credit you may qualify for as low as 5% down with a conventional loan for a 1-Family residence.
As far as giving you credit for the rental income you will be receiving on your Arizona property, you will need to show a lease agreement as well as 1st months rent and security checks deposited into your account from your renter.
If you have any particular questions on Bronx properties I would recommend contacting John Lajara at (917) 848-4655. He is a top producer in his office, which is also #1 in the Bronx. He really knows how to get the job done.