  It was shown recently that associated with a pair of real sequences
$\{\{c_{n}\}_{n=1}^{\infty}, \{d_{n}\}_{n=1}^{\infty}\}$, with
$\{d_{n}\}_{n=1}^{\infty}$ a positive chain sequence, there exists a unique
nontrivial probability measure $\mu$ on the unit circle. The Verblunsky
coefficients $\{\alpha_{n}\}_{n=0}^{\infty}$ associated with the orthogonal
polynomials with respect to $\mu$ are given by the relation $$
\alpha_{n-1}=\overline{\tau}_{n-1}\left[\frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right],
\quad n \geq 1, $$ where $\tau_0 = 1$,
$\tau_{n}=\prod_{k=1}^{n}(1-ic_{k})/(1+ic_{k})$, $n \geq 1$ and
$\{m_{n}\}_{n=0}^{\infty}$ is the minimal parameter sequence of
$\{d_{n}\}_{n=1}^{\infty}$.
  In this manuscript we consider this relation and its consequences by imposing
some restrictions of sign and periodicity on the sequences
$\{c_{n}\}_{n=1}^{\infty}$ and $\{m_{n}\}_{n=1}^{\infty}$. When the sequence $
  \{c_{n}\}_{n=1}^{\infty}$ is of alternating sign, we use information about
the zeros of associated para-orthogonal polynomials to show that there is a gap
in the support of the measure in the neighbourhood of $z= -1$. Furthermore, we
show that it is possible to ge\-nerate periodic Verblunsky coefficients by
choosing periodic sequences $\{c_{n}\}_{n=1}^{\infty}$ and
$\{m_{n}\}_{n=1}^{\infty}$ with the additional restriction $c_{2n}=-c_{2n-1},
\, n\geq 1.$ We also give some results on periodic Verblunsky coefficients from
the point of view of positive chain sequences. An example is provided to
illustrate the results obtained.
