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Asymptotically expand Laplace transform
Assume $\max_{a\leq t\leq b}{\phi (t)} =\phi (a)$, $\phi '(a)\neq 0,f(a)\neq 0$ and $f(t)$ has a Taylor expansion about $t=a$.
Use integration by parts to show that $I(x)=\int_a^b{f(t)e^{x\phi ...
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Approximating the logarithm of a Laplace transform
Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$.
Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e.
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