Let Un be an nn Haar unitary matrix. In this paper, the asymptotic normality and independence of Tr Un, Tr Un2
,..., Tr Unk
are shown by using elementary
methods. More generally, it is shown that the renormalized truncated Haar unitaries
converge to a Gaussian random matrix in distribution.
LetSn be the partial sums of ?-mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(Sn/sn) converges almost surely to ?-88f(x)dF(x). We also obtain strong approximation forH(n)=?k=1nk-1f(Sk/sk)=logn ?-88f(x)dF(x) which will imply the asymptotic normality ofH(n)/log1/2n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ?-mixing random variables.
The problem of random allocation is that of placing
balls independently with equal probability to
boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov  and show that asymptotic normality dominates.
individual observations (see Davison and Hinkley 1997 , p. 71). In any case, because asymptoticnormality still holds, we can rely on a normal approximation adjusted with a bootstrapped variance.
Step 3: Recovering the unconditional