Poisson point process representation
$begingroup$
Let $Pi: ( Omega, mathcal{F}, mathbb{P} ) rightarrow mathbb{R}^d$ be a Poisson point process.
We know that $Pi_0={left | X right |, Xin Pi}$ is a Poisson point process on $mathbb{R}_+$ and that $phi(Pi_0)$ ($phi(r)=v_dr^d, v_d~~ text{is the volume of the unit ball in}~~mathbb{R}^d $) is a Poisson point process on $mathbb{R}_+$ with intensity $l_1$ the Lebesgue measure on $mathbb{R}_+$.
How can we show that there exists a sequence $(X_n)_{ngeq 1}$ of $mathcal{F}$-measurable random variables such that $0< left | X right |_n< left | X right |_{n+1} ,forall ninmathbb{N}^*$ and $ Pi={X_n, ninmathbb{N}^* } ~mathbb{P}-text{as}$. What is the Cumulative distribution function of $left | X_n right |$.
Can someone give a hint?
poisson-process point-processes
asked Jan 21 at 12:49
user636864user636864
11
$endgroup$
add a comment |
$begingroup$
Let $Pi: ( Omega, mathcal{F}, mathbb{P} ) rightarrow mathbb{R}^d$ be a Poisson point process.
We know that $Pi_0={left | X right |, Xin Pi}$ is a Poisson point process on $mathbb{R}_+$ and that $phi(Pi_0)$ ($phi(r)=v_dr^d, v_d~~ text{is the volume of the unit ball in}~~mathbb{R}^d $) is a Poisson point process on $mathbb{R}_+$ with intensity $l_1$ the Lebesgue measure on $mathbb{R}_+$.
How can we show that there exists a sequence $(X_n)_{ngeq 1}$ of $mathcal{F}$-measurable random variables such that $0< left | X right |_n< left | X right |_{n+1} ,forall ninmathbb{N}^*$ and $ Pi={X_n, ninmathbb{N}^* } ~mathbb{P}-text{as}$. What is the Cumulative distribution function of $left | X_n right |$.
Can someone give a hint?
poisson-process point-processes
asked Jan 21 at 12:49
user636864user636864
11
$endgroup$
add a comment |
$begingroup$
Let $Pi: ( Omega, mathcal{F}, mathbb{P} ) rightarrow mathbb{R}^d$ be a Poisson point process.
We know that $Pi_0={left | X right |, Xin Pi}$ is a Poisson point process on $mathbb{R}_+$ and that $phi(Pi_0)$ ($phi(r)=v_dr^d, v_d~~ text{is the volume of the unit ball in}~~mathbb{R}^d $) is a Poisson point process on $mathbb{R}_+$ with intensity $l_1$ the Lebesgue measure on $mathbb{R}_+$.
How can we show that there exists a sequence $(X_n)_{ngeq 1}$ of $mathcal{F}$-measurable random variables such that $0< left | X right |_n< left | X right |_{n+1} ,forall ninmathbb{N}^*$ and $ Pi={X_n, ninmathbb{N}^* } ~mathbb{P}-text{as}$. What is the Cumulative distribution function of $left | X_n right |$.
Can someone give a hint?
poisson-process point-processes
asked Jan 21 at 12:49
user636864user636864
11
$endgroup$
Let $Pi: ( Omega, mathcal{F}, mathbb{P} ) rightarrow mathbb{R}^d$ be a Poisson point process.
We know that $Pi_0={left | X right |, Xin Pi}$ is a Poisson point process on $mathbb{R}_+$ and that $phi(Pi_0)$ ($phi(r)=v_dr^d, v_d~~ text{is the volume of the unit ball in}~~mathbb{R}^d $) is a Poisson point process on $mathbb{R}_+$ with intensity $l_1$ the Lebesgue measure on $mathbb{R}_+$.
How can we show that there exists a sequence $(X_n)_{ngeq 1}$ of $mathcal{F}$-measurable random variables such that $0< left | X right |_n< left | X right |_{n+1} ,forall ninmathbb{N}^*$ and $ Pi={X_n, ninmathbb{N}^* } ~mathbb{P}-text{as}$. What is the Cumulative distribution function of $left | X_n right |$.
Can someone give a hint?
poisson-process point-processes
poisson-process point-processes
asked Jan 21 at 12:49
user636864user636864
11
asked Jan 21 at 12:49
user636864user636864
11
asked Jan 21 at 12:49
user636864user636864
11
asked Jan 21 at 12:49
user636864user636864
11
asked Jan 21 at 12:49
user636864user636864
11
11
add a comment |
add a comment |
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