We can consider the sequence space , which becomes a Polish space (complete, separable metric space) when equipped with the product topology. In the theory of stochastic processes, one is frequently interested in probability measures on this space.
A classical result on measure theory in Polish spaces tells us that any probability measure on a Polish space is tight: given , then there is a compact set such that .
Hence any product probability measure on is tight.
Say we fix an i.i.d. probability measure on . Precisely, for some probability measure on .
Is there a nice, concrete proof that is tight (independent of )?
The reason I ask is tightness is somewhat counter-intuitive here. For instance, if has unbounded support, then, for each , the set is infinite almost surely under (by the 0-1 law). Thinking along those lines, it's not easy to see intuitively.