  We give metric theorems for the property of Borel normality for real numbers
under the assumption of digit dependencies in their expansion in a given
integer base. We quantify precisely how much digit dependence can be allowed
such that, still, almost all real numbers are normal. Our theorem states that
almost all real numbers are normal when at least slightly more than $\log \log
n$ consecutive digits with indices starting at position $n$ are independent. As
the main application, we consider the Toeplitz set $T_P$, which is the set of
all sequences $a_1a_2 \ldots $ of symbols from $\{0, \ldots, b-1\}$ such that
$a_n$ is equal to $a_{pn}$, for every $p$ in $P$ and $n=1,2,\ldots$. Here $b$
is an integer base and $P$ is a finite set of prime numbers. We show that
almost every real number whose base $b$ expansion is in $T_P$ is normal to base
$b$. In the case when $P$ is the singleton set $\{2\}$ we prove that more is
true: almost every real number whose base $b$ expansion is in $T_P$ is normal
to all integer bases. We also consider the Toeplitz transform which maps the
set of all sequences to the set $T_P$ and we characterize the normal sequences
whose Toeplitz transform is normal as well.
