  In this work, we study the properties of a pentadiagonal symmetric matrix
with perturbed corners. More specifically, we present explicit expressions for
characterizing when this matrix is non-negative and positive definite in two
special and important cases. We also give a closed expression for the
determinant of such matrices. Previous works present the determinant in a
recurrence form but not in an explicit one. As an application of these results,
we also study the limiting cumulant generating function associated to the
bivariate sequence of random vectors (n^{-1} (\sum_{k=1}^n X_k^2 , \sum_{k=2}^n
X_k X_{k-1})_{n in N}, when (X_n)_{n in N} is the centered stationary moving
average process of first order with Gaussian innovations. We exhibit the
explicit expression of this limiting cumulant generating function. Finally, we
present three examples illustrating the techniques studied here.
