  This paper deals with the problem of universal lossless coding on a countable
infinite alphabet. It focuses on some classes of sources defined by an envelope
condition on the marginal distribution, namely exponentially decreasing
envelope classes with exponent $\alpha$. The minimax redundancy of
exponentially decreasing envelope classes is proved to be equivalent to
$\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with
a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive
algorithm is provided, whose redundancy is equivalent to the minimax redundancy
