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My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.

More concrete case, let $f:\mathbb{R} \to \mathbb{R}_+$ be a smooth function in $L^1(\mathbb{R})$. If we know that the behavior of its Laplace transform exists on the positive real axis and:

$\int_{\mathbb{R}} f(x) e^{sx} {\rm d}x \geq e^{\frac{s^2}{2}}, \quad \forall s > 0$,

can we conclude that the speed that $f$ converge to zero cannot be faster than $e^{-\frac{x^2}{2}}$, say,

$\liminf_{|x| \to \infty} \frac{f(x)}{e^{\frac{-x^2}{2} (1 - \epsilon)}} > 0$

for some small $\epsilon \in (0, 1)$? In a more probabilistic setup, if we know the moment generating function is lower bounded by that of Gaussian, can we conclude that it is "super-Gaussian"? I know that the other direction seems to be true and is called sub-Gaussian.

If the information on the right-half real axis is not enough, do we need to know more? Will Fourier transform be more helpful? How about the other direction, i.e., lower bound on the Laplace transform and upper bound on the decay of $f$? Thanks.

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# Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.

More concrete case, let $f:\mathbb{R} \to \mathbb{R}_+$ be in $L^1(\mathbb{R})$. If we know that the behavior of its Laplace transform on the positive real axis:

$\int_{\mathbb{R}} f(x) e^{sx} {\rm d}x \geq e^{\frac{s^2}{2}}, \quad \forall s > 0$,

can we conclude that the speed that $f$ converge to zero cannot be faster than $e^{-\frac{x^2}{2}}$, say,

$\liminf_{|x| \to \infty} \frac{f(x)}{e^{\frac{-x^2}{2} (1 - \epsilon)}} > 0$

for some small $\epsilon \in (0, 1)$? In a more probabilistic setup, if we know the moment generating function is lower bounded by that of Gaussian, can we conclude that it is "super-Gaussian"? I know that the other direction seems to be true and is called sub-Gaussian.

If the information on the right-half real axis is not enough, do we need to know more? Will Fourier transform be more helpful? How about the other direction, i.e., lower bound on the Laplace transform and upper bound on the decay of $f$? Thanks.