Let \( P(t) = P(t,i,j) \) be a semigroup of stochastic matrices on the countable set \( I = \{i,j,\dots\}. \) Suppose
\[ \lim_{t\to 0}P(t,i,i) = 1 \quad\text{for each }i \in I .\]
Fix one state \( a \in I \) and abbreviate \( f(t) = P(t,a,a) \).
Suppose \( 0 < \varepsilon < 1 \) and \( f(1) \leq 1 - \varepsilon \). Then
\[ \int_0^1 f(t)\,dt < 1 - \tfrac12 \varepsilon.\]
Suppose \( 0 < \varepsilon < 1/4 \) and \( f(1) \geq 1 - \varepsilon \). Then, for all \( t \) in \( [0,1] \),
\begin{align*} f(t) &\geq \tfrac12[1 + (1 - 4\varepsilon)^{1/2}]\\ & = 1 - \varepsilon - \varepsilon^2 - O(\varepsilon^3). \end{align*}
Suppose \( 0 < \varepsilon < 1/4 \) and \( \int_0^1 f(t)\, dt \geq 1 - \varepsilon \). Then
\[ f(t) > 1 - 2\varepsilon \quad\textit{for }0 \leq t \leq 1 .\]
Note: The last statement can be restated (using algebra) in this more attractive form: if \( 0 < \delta < 1/2 \) and \( f(1) \geq 1 - \delta + \delta^2 \), then \( f(t) \geq 1 - \delta \) for \( 0\leq t \leq 1 \).
