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Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $m=\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $Pr(\{g(t)>c\ :\ \forall t\in A\})=Pr(m>c)$? Or at least what is $Pr(m>0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $Pr(m>c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $Pr(m>0)$.

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $m=\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $Pr(\{g(t)>c\ :\ \forall t\in A\})=Pr(m>c)$? Or at least what is $Pr(m>0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $Pr(m>c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $Pr(m>0)$.

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $m=\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $p(c)=Pr(\{g(t)>c\ :\ \forall t\in A\})=Pr(m>c)$? Or at least what is $p(0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $p(c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $p(0)$.

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $m=\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $Pr(\{g(t)>c\ :\ \forall t\in A\})=Pr(m>c)$? Or at least what is $Pr(m>0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $Pr(m>c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $Pr(m>0)$.

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $p(c)=Pr(\{g(t)>c\ :\ \forall t\in A\})$? Or at least what is $p(0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $p(c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $p(0)$.

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $m=\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $p(c)=Pr(\{g(t)>c\ :\ \forall t\in A\})=Pr(m>c)$? Or at least what is $p(0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike related question I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $p(c)$ behave for small $c$).

EDITED: Per Ylvisaker 1965 we know the expected number of zeros in $A$ is $\frac{T}{\pi}\sqrt{-k''(0)}$. But I couldn't see how it help get $p(0)$.