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For positive claims with finite variance and $c\le \lambda \mu$, the classical risk model has ruin probability one from any finite capital. At claim times, surplus increments are independent copies of $cT-X$ with mean $c/\lambda -\mu$. Negative mean sends their partial sums to minus infinity by the strong law of large numbers. At zero mean, the increments have finite nonzero variance. For every fixed $K$, the central limit theorem gives limiting probability $1/2$ of a partial sum below $-K$. The probability of unboundedness below is therefore at least $1/2$; as a tail event it has probability zero or one by the Kolmogorov zero-one law, and hence one.

For independent and identically distributed random variables with law $N(-1,1)$, $e^{2S_n}$ is a nonnegative martingale, since the increment's moment-generating function equals one at parameter two. The Doob maximal inequality gives $\mathbb{P}(Z>t)\le e^{-2t}$ for the all-time maximum $Z\ge 0$. Integrating the tail gives the displayed bound. For $b\le 0$, $e^{bZ}\le 1$. The strong law of large numbers guarantees finiteness and attainment of the maximum because $S_n\to -\infty$.

For independent unit-rate exponential random variables, the cumulant-generating function is $-\log (1-\theta )$ for $\theta <1$. Its Legendre transform of a cumulant-generating function is $I(x)=x-1-\log x$ for $x>0$ and infinite otherwise. The Cramér theorem gives the upper-tail logarithmic rate $-I(a)$ for $a>1$, and zero for $0\le a\le 1$. At the mean threshold the infima on both the open and closed upper half-lines are zero; below the mean the strong law of large numbers applies.

To estimate ${\mathbb{E}}_f[g(X)]$, sample $Y_1,\dots ,Y_m$ independently from a reference density $h$ whose support covers that of $gf$, and useThe summands have expectation $\int g(x)f(x)dx$, and the strong law of large numbers gives almost-sure convergence under integrability.

If every holding rate of a continuous-time Markov chain is at most $C<\infty$, its holding times can be coupled as $E_n/q_{Y_n}\ge E_n/C$, where the $E_n$ are independent rate-one exponential random variables. The strong law of large numbers gives ${\sum }_n{E}_n=\infty$ almost surely, so the jump times tend to infinity and the chain cannot explode.