Timeline for Integrating Gaussian white noise over a Gaussian density
Current License: CC BY-SA 3.0
10 events
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| Jan 3, 2014 at 18:17 | comment | added | whuber♦ | Matteo, that sum has little to do with the integral you wrote. Because you claim the $X(\theta)$ are independent, treating $\theta$ as a random variable accomplishes nothing at all. Moreover, $\theta$ clearly is not a random variable in your integral formulation. Could you show us where you encountered this integral? | |
| Jan 2, 2014 at 20:58 | comment | added | Matteo Fasiolo | Alecos, yes that was my idea. The problem came about in a simulation: I simulate $\theta_i$ on the real line and I evaluate a function with noise $f(\theta_i) + X(\theta_i)$. I wanted to know what happens when the number of evaluations goes to infinity. | |
| Jan 2, 2014 at 20:52 | comment | added | Matteo Fasiolo | Whuber, I was working on the sum ($\sum_{i=1}^NX(\theta_i)/N$) and I was interested in what happens when $N\to\infty$. I think that the sum has to converge to the expected value of $X(\theta)$, which is zero. I wanted to formalize this fact and I wrote down the integral, but maybe the limit of the sum doesn't correspond to the integral? | |
| Jan 2, 2014 at 20:35 | comment | added | whuber♦ | Matteo, if you are sure the answer is zero, then some part of your description of $X$ must be incorrect. Could you supply a link or reference to where this integral appeared so we could have some context for understanding it? | |
| Jan 2, 2014 at 20:32 | comment | added | Alecos Papadopoulos | So what you are describing by $X(\theta)$ is a stochastic process with continuous and random index? | |
| Jan 2, 2014 at 20:30 | comment | added | Matteo Fasiolo | I am quite sure that Alecos final result is right: the integral is equal to zero. On the other hand, as Whuber is saying, $X(\theta)$ is a stochastic process not a random variable, so I'm not sure about writing $Y = X(\theta)$. I used the notation $X(\theta)$ to mean that for every $\theta$ on the real line there is a random variable. I used the expression "indexed by" to convey the fact that $X(\theta_1), X(\theta_2), \dots$ are iid. | |
| Jan 2, 2014 at 19:46 | comment | added | Alecos Papadopoulos | @whuber I added some discussion. | |
| Jan 2, 2014 at 19:45 | history | edited | Alecos Papadopoulos | CC BY-SA 3.0 |
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| Jan 2, 2014 at 19:23 | comment | added | whuber♦ | I am having trouble making sense of the first two inequalities and the notation. According to the question, $X$ is a family of random variables indexed by $\theta$: it is a stochastic process. Thus the integral is not a Riemann or Lebesgue integral at all and does not appear to be an expectation: it would have to be a random variable. The meaning of $f_\theta(\theta)$ is mysterious. Could you please explain your interpretation, explain your notation, and justify your steps? | |
| Jan 2, 2014 at 18:59 | history | answered | Alecos Papadopoulos | CC BY-SA 3.0 |