We show that the orthogonal projection operator onto the range of the adjoint T⁎ of a linear operator T can be represented as UT, where U is an invertible linear operator. Given a Normal random vector Y and a linear operator T, we use this representation to obtain a linear operator trueTˆ such that trueTˆY is independent of TY and Y−trueTˆY is an affine function of TY. We then use this decomposition to prove that the conditional distribution of a Normal random vector Y given scriptTY, where scriptT is a linear transformation, is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional Normal random vector, where k

The asymptotic solution to the problem of comparing the means of two heteroscedastic populations, based on two random samples from the populations, hinges on the pivot underpinning the construction of the confidence interval and the test statistic being asymptotically standard Normal, which is known to happen if the two samples are independent and the ratio of the sample sizes converges to a finite positive number. This restriction on the asymptotic behavior of the ratio of the sample sizes carries the risk of rendering the asymptotic justification of the finite sample approximation invalid. It turns out that neither the restriction on the asymptotic behavior of the ratio of the sample sizes nor the assumption of cross sample independence is necessary for the pivotal convergence in question to take place. If the joint distribution of the standardized sample means converges to a spherically symmetric distribution, then that distribution must be bivariate standard Normal (which can happen without the assumption of cross sample independence), and the aforesaid pivotal convergence holds.! " # ! . .MSC 2010 subject classifications: 62E20, 62G20.

We consider three matrix models of order 2 with one random entry and the other three entries being deterministic. In the first model, we let ∼ Bernoulli 1 2. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when ∼ Bernoulli (p) and p ∈ [0, 1] is a parameter. Both of

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