  The main theme of this paper is the enumeration of the occurrence of a
pattern in words and permutations. We mainly focus on asymptotic properties of
the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a
given pattern $v$ exactly $r$ times. In addition, we study the asymptotic
behavior of the random variable $X_n,$ the number of pattern occurrences in a
random $n$-array word. The two topics are closely related through the identity
$P(X_n=r) = $ $\frac{1}{k^n}f_r^v(k,n).$ In particular, we show that for any
$r\geq 0,$ the Stanley-Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{1/n}$ converges
to a limit independent of $r,$ and determine the value of the limit. We then
obtain several limit theorems for the distribution of $X_n,$ including a CLT,
large deviation estimates, and the exact growth rate of the entropy of $X_n.$
Furthermore, we introduce a concept of weak avoidance and link it to a certain
family of non-product measures on words that penalize pattern occurrences but
do not forbid them entirely. We analyze this family of probability measures in
a small parameter regime, where the distributions can be understood as a
perturbation of a uniform measure. Finally, we extend some of our results for
words, including the one regarding the equivalence of the limits of the
Stanley-Wilf sequences, to pattern occurrences in permutations.
