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We formalize the tail redundancy of a collection of distributions over acountably infinite alphabet, and show that this fundamental quantitycharacterizes the asymptotic per-symbol redundancy of universally compressingsequences generated iid from a collection $\mathcal P$ of distributions over acountably infinite alphabet. Contrary to the worst case formulations ofuniversal compression, finite single letter (average case) redundancy of$\mathcal P$ does not automatically imply that the expected redundancy ofdescribing length-$n$ strings sampled iid from $\mathcal P$ grows sublinearlywith $n$. Instead, we prove that universal compression of length-$n$ \iidsequences from $\mathcal P$ is characterized by how well the tails ofdistributions in $\mathcal P$ can be universally described, showing that theasymptotic per-symbol redundancy of iid strings is equal to the tailredundancy.

Tail redundancy and its characterization of compression of memoryless sources