  In this paper we study two one-parameter families of random band Toeplitz
matrices: \[
A_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}\delta_{|i-j|\le[b_nt]}\Big)_{i,j=1}^n
\quad\text{and}\quad B_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}(t)\delta_{|i-j|\le
b_n}\Big)_{i,j=1}^n \] where
  1. $a_0=0$, $\{a_1,a_2,..\}$ in $A_n(t)$ are independent random variables and
$a_{-i}=a_i$
  2. $a_0(t)=0$, $\{a_1(t),a_2(t),...\}$ in $B_n(t)$ are independent copies of
the standard Brownian motion at time $t$ and $a_{-i}(t)=a_i(t)$.
  As $t$ varies, the empirical measures $\mu(A_n(t))$ and $\mu(B_n(t))$ are
measure valued stochastic processes. The purpose of this paper is to study the
fluctuations of $\mu(A_n(t))$ and $\mu(B_n(t))$ as $n$ goes to $\infty$. Given
a monomial $f(x)=x^p$ with $p\ge2$, the corresponding rescaled fluctuations of
$\mu(A_n(t))$ and $\mu(B_n(t))$ are \[\sqrt{b_n}\Big(\int
f(x)d\mu(A_n(t))-E[\int
f(x)d\mu(A_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(A_n(t)^p)-E[\text{tr}(A_n(t)^p)]\Big),
\quad(1)\] \[\sqrt{b_n}\Big(\int f(x)d\mu(B_n(t))-E[\int
f(x)d\mu(B_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(B_n(t)^p)-E[\text{tr}(B_n(t)^p)]\Big)
\quad(2)\] respectively. We will prove that (1) and (2) converge to centered
Gaussian families $\{Z_p(t)\}$ and $\{W_p(t)\}$ respectively. The covariance
structure $E[Z_p(t_1)Z_q(t_2)]$ and $E[W_p(t_1)W_q(t_2)]$ are obtained for all
$p,q\ge 2$; $t_1,t_2\ge 0,$ and are both homogeneous polynomials of $t_1$ and
$t_2$ for fixed $p,q$. In particular, $Z_2(t)$ is the Brownian motion and
$Z_3(t)$ is the same as $W_2(t)$ up to a constant.
  The main method of this paper is the moment method.
