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Is it possible to become a tenure-track assistant professor of statistics department directly after the PhD, without having to do a postdoc? I'm interested in the top schools in the US (i.e., the Ivy League).
In Canada, a good PhD graduate from a statistics program would go into academia without having to do the postdoc, but I am not sure what the situation is like in the US.
It is possible to get a faculty job without a postdoc - in fact it's possible to get a faculty job without a PhD (see Noah Snyder's answer to a related question).
Of course, what is possible and what is likely can be completely different. It's possible to be dealt a royal flush in poker but not at all likely. There are hundreds of thousands of professors worldwide and chances are the number who haven't done a postdoc can be counted on two hands. Since you're further restricting the sample to the top schools, well, let's just say that it's about as likely as being dealt a royal flush.
