Mixing
Distribution of an intersection between a Poisson point process and a random set
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Let $D$ be a Poisson Point Process on $mathbbR$ of rate $lambda > 0$. Suppose $(X_t)_t geq 0$ is a continuous time Markov Chain taking values $0,1$. The chain and the point process are independent. Now define:
$Z = int_0^1 1_X_t = 1 dt $
I wonder if it's true that:
$|D cap t :X_t = 1, t in [0,1]| sim |D cap [0,Z]| $
I guess it is true and one way I tried to approach this problem is to condition both events $D cap t :X_t = 1, t in [0,1]$ and $D cap [0,Z]$ on the jumping times $tau_i_i geq 0$ of the chain. But I can show the equality only conditioned on the times assuming specific values, i.e. $tau_i=r_i ; ; forall igeq0$ for some $r_i in mathbbR$, which I guess it's not enough. Any thoughts on how I can go from this to prove claim?
probability stochastic-processes
asked Aug 1 at 11:01
Flamematico
62
add a comment |Â
up vote
1
down vote
favorite
Let $D$ be a Poisson Point Process on $mathbbR$ of rate $lambda > 0$. Suppose $(X_t)_t geq 0$ is a continuous time Markov Chain taking values $0,1$. The chain and the point process are independent. Now define:
$Z = int_0^1 1_X_t = 1 dt $
I wonder if it's true that:
$|D cap t :X_t = 1, t in [0,1]| sim |D cap [0,Z]| $
I guess it is true and one way I tried to approach this problem is to condition both events $D cap t :X_t = 1, t in [0,1]$ and $D cap [0,Z]$ on the jumping times $tau_i_i geq 0$ of the chain. But I can show the equality only conditioned on the times assuming specific values, i.e. $tau_i=r_i ; ; forall igeq0$ for some $r_i in mathbbR$, which I guess it's not enough. Any thoughts on how I can go from this to prove claim?
probability stochastic-processes
asked Aug 1 at 11:01
Flamematico
62
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
1
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $D$ be a Poisson Point Process on $mathbbR$ of rate $lambda > 0$. Suppose $(X_t)_t geq 0$ is a continuous time Markov Chain taking values $0,1$. The chain and the point process are independent. Now define:
$Z = int_0^1 1_X_t = 1 dt $
I wonder if it's true that:
$|D cap t :X_t = 1, t in [0,1]| sim |D cap [0,Z]| $
I guess it is true and one way I tried to approach this problem is to condition both events $D cap t :X_t = 1, t in [0,1]$ and $D cap [0,Z]$ on the jumping times $tau_i_i geq 0$ of the chain. But I can show the equality only conditioned on the times assuming specific values, i.e. $tau_i=r_i ; ; forall igeq0$ for some $r_i in mathbbR$, which I guess it's not enough. Any thoughts on how I can go from this to prove claim?
probability stochastic-processes
asked Aug 1 at 11:01
Flamematico
62
Let $D$ be a Poisson Point Process on $mathbbR$ of rate $lambda > 0$. Suppose $(X_t)_t geq 0$ is a continuous time Markov Chain taking values $0,1$. The chain and the point process are independent. Now define:
$Z = int_0^1 1_X_t = 1 dt $
I wonder if it's true that:
$|D cap t :X_t = 1, t in [0,1]| sim |D cap [0,Z]| $
I guess it is true and one way I tried to approach this problem is to condition both events $D cap t :X_t = 1, t in [0,1]$ and $D cap [0,Z]$ on the jumping times $tau_i_i geq 0$ of the chain. But I can show the equality only conditioned on the times assuming specific values, i.e. $tau_i=r_i ; ; forall igeq0$ for some $r_i in mathbbR$, which I guess it's not enough. Any thoughts on how I can go from this to prove claim?
probability stochastic-processes
asked Aug 1 at 11:01
Flamematico
62
edited Aug 1 at 14:53
asked Aug 1 at 11:01
Flamematico
62
asked Aug 1 at 11:01
Flamematico
62
asked Aug 1 at 11:01
Flamematico
62
62
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
1
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39
add a comment |Â
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
1
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
1
1
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2868951%2fdistribution-of-an-intersection-between-a-poisson-point-process-and-a-random-set%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
You have not fully specified the Markov Chain. If you were to give the associated probability matrix we are able to answer this question.
– Jan
Aug 1 at 13:42
At first there shouldn't be any hypothesis on the transitions probabilities.
– Flamematico
Aug 1 at 14:04
1
The notation $D cap [0,Z]$ is weird since $D$ is not a set but a process. Therefore it is not clear to me exactly what you mean. Do however note that Note that if we have $X_t=1$ on $t in [0,frac13]$ and on $[frac23,1]$ and we have $X_t=0$ on $t in (frac13, frac23)$ we will have $Z=frac23$ so $[0,Z]=[0, frac23]$,while $ t :X_t = 1=[0,frac13]cup [frac23,1]$ so at least we know $[0,Z] neq t :X_t = 1$ in general.
– Jan
Aug 1 at 14:19
Thanks form commenting that! I realized that I should be talking about the distribution of the cardinality of the sets, and the question is now edited. But regarding your comment, a Poisson Point Process $D$ can be viewed as a random discrete subset of the real line, with the property that, for any Lebesgue measurable $A subset mathbbR$ we have $mathbbP(|D cap A| = k) = frac[lambda l(A)]^kk!e^-lambda l(A)$ where $lambda > 0$ is the intensity of the process and $l$ is the Lebesgue measure. So, in your example, the process would be identically distributed.
– Flamematico
Aug 1 at 14:39