  In this work, we consider symmetric random Toeplitz matrices $T_n$ generated
by i.i.d. zero mean random variables ${X_k}$ satisfying the moment conditions:
$E|X_k|^2=1$ and $\E|X_1|^n \le n^{\sqrt{n}}$ for all $n\ge 3$. We prove that
the largest eigenvalue of $T_n$ scaled by $\sqrt{n log(n)}$ converges almost
surely to $1$.
