# Density of integral of exponential of Gaussian bailiwick with log-covariance retort

Hello expensive customer to our community We will proffer you an answer to this query Density of integral of exponential of Gaussian bailiwick with log-covariance ,and the retort will breathe typical via documented data sources, We welcome you and proffer you contemporary questions and solutions, Many customer are questioning in regards to the retort to this query.

## Density of integral of exponential of Gaussian bailiwick with log-covariance

Here we make use of the Gaussian bailiwick $$Vin N(0,ln(delta/epsilon))$$ over $$[0,1]$$ with covariance
$$commence{equation} mathbb{E}[V(x_{1} )V(x_{2} ) ]=ln(frac{delta}{|x_{2}-x_{1}|vee epsilon})=left{commence{matrix} ln(frac{delta}{epsilon} ) &, |x_{2}-x_{1}|leq epsilon ln(frac{delta}{x_{2}-x_{1}}) &, deltageq |x_{2}-x_{1}|geq epsilon 0&,|x_{2}-x_{1}|>delta stop{matrix}capable.. stop{equation}$$

assume the touchstone $$mu(x)=int_{0}^{x}e^{barrier{V}(s)}ds$$, the place $$barrier{V}(s)=V(s)-frac{1}{2}E(V^{2}(s))$$ (in order that $$E[mu(x)]=x$$).

Q: We attempt to appraise the likelihood
$$P[aleq mu(1)leq b]$$
for common $$0 (or aircraft higher the density obligation of $$mu$$).

Some feedback from the literature

• In the paper On the density capabilities of integrals of Gaussian random fields, they hunt the tail $$F^{c}(a)=P[aleq mu(1)]$$ for sizable $$a$$. (they too array that the density $$f(a)=F'(a)$$ exists and has Gaussian tail).

• In the papers On the coterie, GMC…, The distribution of Gaussian multiplicative chaos on the unit interval, they handle to compute the density within the illustration $$epsilon=0$$ above.

• In The Integral of Geometric Brownian Motion, he computes the density for $$int_{0}^{x}e^{2B_{t}+2mu t}dt$$, the place $$B_{t}$$ is Brownian movement and $$muin mathbb{R}$$ is a ceaseless.

One workable route to accumulate some estimates is through the use of the Borel-Sudakov appraise
$$P[sup_{sin [0,h]} V(s)-V(0)geq u]lesssim h e^{-u^{2}/2ln(delta/epsilon)}$$
and changing the above by
$$P[frac{a}{e^{u_{1}}}leq e^{bar{V}(0)}leq frac{b}{e^{-u_{2}}}]+P[sup_{sin [0,1]} V(s)-V(0)geq u_{1}]+P[inf_{sin [0,1]} V(s)-V(0)geq u_{2}].$$

So when $$a,b$$ are each sizable, one can purchase some estimates (as instructed in 1 too). Anyhow given the technicalities in 1, we doubt there may be any concrete density, we at the least await to accumulate varied estimates for diminutive and immense $$a,b$$.

we’ll proffer you the answer to Density of integral of exponential of Gaussian bailiwick with log-covariance query through our community which brings all of the solutions from a number of reliable sources.