It happened about 1:30 p.m. Thursday in the parking lot of the Schaumburg mall.
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An armed male suspect took a purse, Schaumburg officials said.
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Oak Brook police said they later found the Mazda suspected of being involved near Oakbrook Center.
"It happens quite often where they hit one mall and they go to the next one," Oak Brook Police Department Deputy Chief Reid Foltyniewicz said. "The next nearest one is us, so we had officers head that way."
Officers said the driver rammed into a police car while they pursued him.
The suspect was eventually arrested on Interstate 294 at Ogden Avenue, and a gun was recovered about 2:20 p.m.
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The DuPage County State's Attorney's Office said the loaded gun was thrown out of the car's window during the chase.
Downers Grove, Elmhurst and Hinsdale police also helped stop the vehicle.
The mother and baby were not hurt.
There was a large police presence at Oak Brook mall during the incident.
The state's attorney's office said Friday that Jaquan Wilkins, 28, of Chicago was charged with illegal possession of a firearm, criminal damage to state-supported property, aggravated fleeing and eluding and aggravated unlawful use of a weapon in connection with the incident.
"This was a violent offender creating a tremendous danger to the public," DuPage County State's Attorney Robert Berlin said.
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Police said the technology helped them make an arrest quickly before the the suspect could commit more crimes.
"Now we know the direction they're going and now we can get officers to look ahead instead of going to the scene five or ten minutes late, we can look ahead," Foltyniewicz said.
Wilkins appeared in court Friday morning, and was denied pre-trial release, the state's attorney's office said.
He was out on bond for Will County attempted murder and aggravated battery charges, DuPage County officials said.
Wilkins is next due in court Dec. 4.