  In this paper, we discuss the distribution of the t-statistic under the
assumption of normal autoregressive distribution for the underlying discrete
time process. This result generalizes the classical result of the traditional
t-distribution where the underlying discrete time process follows an
uncorrelated normal distribution. However, for AR(1), the underlying process is
correlated. All traditional results break down and the resulting t-statistic is
a new distribution that converges asymptotically to a normal. We give an
explicit formula for this new distribution obtained as the ratio of two
dependent distribution (a normal and the distribution of the norm of another
independent normal distribution). We also provide a modified statistic that
follows a non central t-distribution. Its derivation comes from finding an
orthogonal basis for the the initial circulant Toeplitz covariance matrix. Our
findings are consistent with the asymptotic distribution for the t-statistic
derived for the asympotic case of large number of observations or zero
correlation. This exact finding of this distribution has applications in
multiple fields and in particular provides a way to derive the exact
distribution of the Sharpe ratio under normal AR(1) assumptions.
