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Tierney moved from Celtic to Arsenal for a reported fee of £25 million on transfer deadline day and will be in competition with Sead Kolasinac and Nacho Monreal for a starting spot once fit.
His former team-mate Brown believes the 22-year-old Scotland international has what it takes to play in the upper echelons of European football.
"You don’t know what could happen,” Brown told The Herald.
He could have a fantastic couple of seasons and end up at Barcelona or PSG or wherever.
Brown also insisted Tierney would be welcome back to Celtic Park and says there is no bitterness towards the left-back for leaving.
“There are pictures on the wall all over the club of players who have left, and everyone would wish them all the best. They can come back as well, that’s what this club is all about," he added.
Tierney (L) played alongside Brown (R) at Celtic for five seasons
“I see Ajax bringing players through and selling them on - and they have a party for them leaving. We try and criticise. It’s a different level.
“We wish him all the best. We just concentrate on now and [new left back Boli Bolingoli] has to push on and try to get to the levels that Kieran did.”
Chris Froome (Stage 2, 2015)