In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using time-sharing to the strongest user, dirty paper coding (DPC), and beamforming when the number of users (receivers) n is large. Throughout the paper, we assume a fix av-erage transmit power and consider a block fading Rayleigh channel. First, we show that for a system with M transmit antennas and users equipped with N antennas, the sum rate scales like M log log nN for DPC and beamforming when M is fixed and for any N (either growing to in-finity or not). On the other hand, when both M and N are fixed, the sum rate of time-sharing to the strongest user scales like min(M, N) log log n. Therefore, the asymptotic gain of DPC over time-sharing for the sum rate is M min(M,N) when M and N are fixed. It is also shown that if M grows as log n, the sum rate of DPC and beamforming will grow linearly in M, but with different constant multiplicative factors. In this region, the sum rate capacity of time-sharing scales like N log log n.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z |> c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z) | is of order |z | a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.

This research was supported by the Office of Naval Research under Contract No. Nonr-855(06) for resear~h in probability and statistics at Chapel Hill. Reproduction in whole or in part is permitted for any purpose

We investigate the role of market transparency in repeated first-price auctions. We consider a setting with private and independent values across bidders. The values are assumed to be perfectly persistent over time. We analyze the first-price auction under three distinct disclosure regimes regarding the bid and award history. Of particular interest is the minimal disclosure regime, in which each bidder only learns privately whether he won or lost the auction at the end of each round. In equilibrium, the winner of the initial auction lowers his bids over time, while losers keep their bids constant, in anticipation of the winner’s lower future bids. This equilibrium is efficient, and all information is eventually revealed. Importantly, this disclosure regime does not give rise to pooling equilibria. We contrast the minimal disclosure setting with the case in which all bids are public, and the case in which only the winner’s bids are public. In these settings, an inefficient pooling equilibrium with low revenues always exists with a sufficiently large number of bidders.