  The objective of this manuscript is to study directly the Favard type theorem
associated with the three term recurrence formula % \[ R_{n+1}(z) =
\big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1} z R_{n-1}(z), \quad n
\geq 1, \] % with $R_{0}(z) =1$ and $R_{1}(z) = (1+ic_{1})z+(1-ic_{1})$, where
$\{c_n\}_{n=1}^{\infty}$ is a real sequence and $\{d_n\}_{n=1}^{\infty}$ is a
positive chain sequence. We establish that there exists an unique nontrivial
probability measure $\mu$ on the unit circle for which $\{R_n(z) -
2(1-m_n)R_{n-1}(z)\}$ gives the sequence of orthogonal polynomials. Here,
$\{m_n\}_{n=0}^{\infty}$ is the minimal parameter sequence of the positive
chain sequence $\{d_n\}_{n=1}^{\infty}$. The element $d_1$ of the chain
sequence, which does not effect the polynomials $R_n$, has an influence in the
derived probability measure $\mu$ and hence, in the associated orthogonal
polynomials on the unit circle. To be precise, if $\{M_n\}_{n=0}^{\infty}$ is
the maximal parameter sequence of the chain sequence, then the measure $\mu$ is
such that $M_0$ is the size of its mass at $z=1$. An example is also provided
to completely illustrates the results obtained.
