Stein's method and the multivariate CLT for traces of powers on the classical compact groups

Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector (\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d)) tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.Comment: Main result strengthened, proof of Prop. 1.3 adde

Banks' regulatory capital buffer and the business cycle: evidence for German savings and cooperative banks

This paper analyzes the effect of the business cycle on the regulatory capital buffer of German savings and cooperative banks in the period 1993-2003. The capital buffer is found to fluctuate anticyclically over the business cycle. The fluctuation is stronger for savings banks than for cooperative banks, as, for savings banks, risk-weighted assets fluctuate more strongly with the business cycle. Further, low-capitalized banks do not catch up with their wellcapitalized peers. The gap between low-capitalized and well capitalized banks even widened over the observation period. Finally, low-capitalized banks do not decrease risk-weighted assets in a business cycle downturn by more than well-capitalized banks. This finding seems to imply that their low capitalization does not force them to retreat from lending. --Capital Regulation,Bank Capital,Business Cycle Fluctuations