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 __\( 0\leq t \leq 1 \)__.