Let $A$ be a finite set. Let $M : A \times A \to {\bf R}$ be a symmetric function, which is positive-semidefinite when regarded as an $A \times A$ matrix.
Let $P(A)$ be the set of vectors $p = (p_a)_{a \in A} \in {\bf R}^A$ such that for all $a \in A$, $p_a \ge 0$, and $\sum_{a \in A} p_a = 1$, the set of "probability distributions on A"
Then let $S(p) = \sum_{(a,b) \in A \times A} M(a,b) p_a p_b$, and $U_t = \{p \in P(a) : S(p) \leq r\}$.
Let $t_{\rm min}$ be the infimum of $\{t \in {\bf R} : U_t \neq \emptyset \}$, $\mu$ be any element of $P(A)$ whatsoever, and $t_0 = S(\mu)$.
If $t_{\rm min} < t < t_0$, and $X$ is a random string in $A^n$ where each letter has probability of occurring at each position corresponding to those in $\mu$, then we want to estimate the probability of that the letter frequencies in $X$ correspond to a vector in $U_t$.
In particular I think it can be written as a function of the form $e^{-cn + o(n)}$ for some constant $c > 0$. [Little o-notation]