Dgdrxrt

How easy is it to start Magic from scratch?

Large drywall patch supports

What does "I’d sit this one out, Cap," imply or mean in the context?

Is there a problem with hiding "forgot password" until it's needed?

Would a high gravity rocky planet be guaranteed to have an atmosphere?

Term for the "extreme-extension" version of a straw man fallacy?

Inappropriate reference requests from Journal reviewers

Is a stroke of luck acceptable after a series of unfavorable events?

Short story about space worker geeks who zone out by 'listening' to radiation from stars

Gears on left are inverse to gears on right?

Escape a backup date in a file name

Why Were Madagascar and New Zealand Discovered So Late?

Detecting if an element is found inside a container

How does buying out courses with grant money work?

How long to clear the 'suck zone' of a turbofan after start is initiated?

Do sorcerers' Subtle Spells require a skill check to be unseen?

How to Reset Passwords on Multiple Websites Easily?

Sort a list by elements of another list

How to safely derail a train during transit?

Why are there no referendums in the US?

Was Spock the First Vulcan in Starfleet?

Method to test if a number is a perfect power?

Is the destination of a commercial flight important for the pilot?

How to run a prison with the smallest amount of guards?

Laplace functional for given process.

“Paradox” of a Poisson process on $mathbb R$Spectral process for the Ornstein-Uhlenbeck processDistribution of stochastic integral w.r.t. to centered Poisson processConditional distribution of compounded Poisson processProve that a process has independent incrementsDistribution of the increments of a Compound Poisson processCompute $Eleft[fracW_N(t)+1N(t) + 1right]$ where $(N(t))$ is a homogenous Poisson processHow to compute the integral of a renewal process?How to construct a Poisson process not based on Lebesgue measure?Compound Binomial - Exponential process

0

$begingroup$

Consider $X_1 dots $ i.r.v with $Exp(lambda)$ distribution. Let's consider $S_n = X_1 + dots + X_n$.

So we may construct a point process $Y = Y(B) = sum mathbb1_B(S_n) , Bin mathfrakB(mathbbR) $.

No we need to find $mathfrakL(f) = mathbbE(exp[-sum_n f(S_n)])$,

My attempt :

First of all let's suppose that we have $xi $~$Gamma(n,frac1lambda)$ , when $mathbbE(exp[-sxi]) = left(dfraclambdalambda + sright)^n$.

Let's consider $f = sum_k = 1^m a_kmathbb1_B_k$, hence we have :

$mathfrakL(f) = mathbbEexp[-sum_n^infty sum_k^m a_k mathbb1_B_k(S_n)]$ replace sum sign due to positive terms.

$mathfrakL(f) = prod mathbbE exp[-a_k Y(B_k)]$, but there is the stuck problem.

What should I do now?

probability stochastic-processes expected-value

share|cite|improve this question

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

$endgroup$

This question has an open bounty worth +100
reputation from openspace ending ending at 2019-04-03 07:12:38Z">in 6 days.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the domain of f? Is it a generic measurable real function? And where the definition of Y enters in the question?
  $endgroup$
  – Thomas
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas $f$ should be a Borel-function
  $endgroup$
  – openspace
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas I mean Borel-measurable function
  $endgroup$
  – openspace
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx. What role instead plays Y? It seems that it is not necessary to define the laplace functional? I am asking to be sure that I am getting the problem...
  $endgroup$
  – Thomas
  18 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas Y is a point process.
  $endgroup$
  – openspace
  16 hours ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Consider $X_1 dots $ i.r.v with $Exp(lambda)$ distribution. Let's consider $S_n = X_1 + dots + X_n$.

So we may construct a point process $Y = Y(B) = sum mathbb1_B(S_n) , Bin mathfrakB(mathbbR) $.

No we need to find $mathfrakL(f) = mathbbE(exp[-sum_n f(S_n)])$,

My attempt :

First of all let's suppose that we have $xi $~$Gamma(n,frac1lambda)$ , when $mathbbE(exp[-sxi]) = left(dfraclambdalambda + sright)^n$.

Let's consider $f = sum_k = 1^m a_kmathbb1_B_k$, hence we have :

$mathfrakL(f) = mathbbEexp[-sum_n^infty sum_k^m a_k mathbb1_B_k(S_n)]$ replace sum sign due to positive terms.

$mathfrakL(f) = prod mathbbE exp[-a_k Y(B_k)]$, but there is the stuck problem.

What should I do now?

probability stochastic-processes expected-value

share|cite|improve this question

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

$endgroup$

This question has an open bounty worth +100
reputation from openspace ending ending at 2019-04-03 07:12:38Z">in 6 days.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the domain of f? Is it a generic measurable real function? And where the definition of Y enters in the question?
  $endgroup$
  – Thomas
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas $f$ should be a Borel-function
  $endgroup$
  – openspace
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas I mean Borel-measurable function
  $endgroup$
  – openspace
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx. What role instead plays Y? It seems that it is not necessary to define the laplace functional? I am asking to be sure that I am getting the problem...
  $endgroup$
  – Thomas
  18 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Thomas Y is a point process.
  $endgroup$
  – openspace
  16 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Consider $X_1 dots $ i.r.v with $Exp(lambda)$ distribution. Let's consider $S_n = X_1 + dots + X_n$.

So we may construct a point process $Y = Y(B) = sum mathbb1_B(S_n) , Bin mathfrakB(mathbbR) $.

No we need to find $mathfrakL(f) = mathbbE(exp[-sum_n f(S_n)])$,

My attempt :

First of all let's suppose that we have $xi $~$Gamma(n,frac1lambda)$ , when $mathbbE(exp[-sxi]) = left(dfraclambdalambda + sright)^n$.

Let's consider $f = sum_k = 1^m a_kmathbb1_B_k$, hence we have :

$mathfrakL(f) = mathbbEexp[-sum_n^infty sum_k^m a_k mathbb1_B_k(S_n)]$ replace sum sign due to positive terms.

$mathfrakL(f) = prod mathbbE exp[-a_k Y(B_k)]$, but there is the stuck problem.

What should I do now?

probability stochastic-processes expected-value

share|cite|improve this question

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

$endgroup$

share|cite|improve this question

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

share|cite|improve this question

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

asked Mar 17 at 21:31

openspaceopenspace

3,3512822

asked Mar 17 at 21:31

openspaceopenspace

3,3512822