Mario Edward Garnett was living in Oklahoma City at the time of the arrest, Secret Service spokesman Max Millien said. He would not say how the case was resolved.
Authorities have connected Garnett to a Dec. 23 bank robbery attempt in Atlanta, a bank robbery later that same day in Tupelo, Miss., and the shooting death of one officer and the wounding of another just after the Tupelo robbery.
A Phoenix detective shot and killed Garnett just after he robbed a bank Saturday morning.
Phoenix police spokesman James Holmes said at a Sunday afternoon news conference that the 40-year-old Garnett was from the Midwest, had served for four years in the U.S. Army and appeared to have no friends or relatives in Arizona.
Earlier Sunday, the FBI's special agent in charge in Jackson, Miss., said that records showed the suspect's cellphone was in Atlanta, Phoenix and Tupelo at the time of each incident.
The agent, Daniel McMullen, also told reporters in in Tupelo that there were "numerous similarities" between the three bank robberies, including the clothing worn by the suspect, statements he made during the incidents and his "overall modus operandi.
Surveillance photos from the attempted bank robbery in Atlanta showed a man wearing a black-and-white patterned jacket tried to rob a Bank of America branch around 9:30 a.m. on Dec. 23. He was wearing a black ski mask and black gloves and carrying a handgun.
After failing to get any money from a teller, the man demanded and received an undisclosed amount of money from someone who was using an ATM in the bank's lobby. The man was seen leaving in a light gray car with no license plate.
Later that same day, the suspect stole an undisclosed amount of money from the BancorpSouth in Tupelo before fleeing in a gray sedan, the FBI has said.
Officers Gale Stauffer and Joseph Maher were responding to a bank alarm and a report that the suspect had fled in a white SUV when they spotted such a vehicle stalled in traffic. The officers were confronting the SUV's driver when the suspect exited the gray sedan, which was stopped behind the SUV, and ambushed the officers, according to McMullen.
Stauffer died and was buried Friday after a funeral attended by as many as 1,000 people. Dozens of uniformed officers from Mississippi, Alabama, Tennessee and elsewhere attended the service.
Stauffer's wife, Beth, said her family was grateful for investigators' work and for the support they've received.
"We can truly begin the healing process, and that could not have happened without the hard work of so very many," she told reporters Sunday. "Gale would be so proud."
Tupelo police Chief Bart Aguirre told the Northeast Mississippi Daily Journal that Maher's condition has improved and he was able to walk some with assistance Friday. Maher moved out of the intensive care unit at North Mississippi Medical Center and into a private room.
The Phoenix shooting happened Saturday morning at a Compass Bank branch in the near-northwest part of the city after someone called 911 to report seeing a man run into the bank while wearing a mask.
There were five people inside the bank during the robbery, Holmes said. A customer who ran outside when Garnett turned his attention to the bank manager also called 911.
The first officer to arrive saw the man leave the bank with a bag and a handgun.
"The officer drew his service weapon and began giving the suspect commands at which time the suspect raised his weapon and began firing at the officer," Holmes said in an earlier statement.
A detective who had just arrived at the scene shot the suspect, who was pronounced dead after being taken to a Phoenix hospital. No officers or bank employees were injured.
Holmes said Garnett drove a rental car to the Phoenix bank. It was unclear where Garnett had been and what he had been doing while in the state.
"We have no idea what he did between the 23rd of December and the 28th of December," he said.
Associated Press writer Michael Kunzelman in New Orleans and AP Radio reporter Jackie Quinn in Washington contributed to this report.