  We analyze the asymptotic average and effective throughputs of a multiuser
diversity scheme for a secondary multiuser network consisting of multiple
secondary users (transmitters) and one secondary receiver. Considering a
transmit power adaptation strategy at the secondary users to satisfy the
instantaneous interference constraint at the primary receiver, the secondary
receiver selects the $k$-th best secondary user for transmission, namely, the
one with the $k$-th highest signal-to-noise ratio (SNR). We use extreme value
theory to show that the $k$-th highest SNR converges uniformly in distribution
to an inverse gamma random variable for a fixed $k$ and large number of
secondary users. We use this result to derive closed-form asymptotic
expressions for the average and effective throughputs of the $k$-th best
secondary user.
