  The Clarkson-McLeod solutions of the fourth Painlev\'e equation behave like
$\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where
$\kappa$ is some real constant and $D_{\alpha-\frac{1}{2}}(x)$ is the parabolic
cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we
derive the asymptotic behaviors for this class of solutions as $x\to-\infty$.
This completes a proof of Clarkson and McLeod's conjecture on the asymptotics
of this family of solutions. The total integrals of the Clarkson-McLeod
solutions and the asymptotic approximations of the $\sigma$-form of this family
of solutions are also derived. Furthermore, we find a determinantal
representation of the $\sigma$-form of the Clarkson-McLeod solutions via an
integrable operator with the parabolic cylinder kernel.
