If the universe is infinite couldn't there have been several "big bangs?" We know that all that we call the universe existed within time-zero billions of years ago, but what about further out into space? What about further than we can ever imagine? Couldn't some galaxies far out be the result of some other event besides our "big bang?" I'm not talking about other universes in other dimensions, I'm just saying "how do we know that all the stuff in infinite space came from the same place?" BTW couldn't there have been a "big bang" that happened billions of years before what we call the "big bang" that made a previous known universe that collapsed back into time-zero only to explode again? Friedrich Nietzsche seemed to believe in eternal reoccurrence, but I don't know if he thought about it on a cosmic level.
This becomes a matter of terminology as much as anything else.
General relativity, which is the underlying theory for the Big Bang, does not allow situations like you describe. And the current observations, including red-shifts, are all consistent with GR.
On the other hand, what you describe is, essentially, what happens in certain multiverse models.