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Apr
13
revised Could we define two random variables such that the product of them is Normal distribution(Gaussian)?
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Apr
13
suggested approved edit on Could we define two random variables such that the product of them is Normal distribution(Gaussian)?
Apr
3
revised Conditional Probability in Geometric Distribution
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Apr
3
suggested approved edit on Conditional Probability in Geometric Distribution
Apr
1
comment If $\mu$ is $f$-invariant for a collection generating the Borel $\sigma$-algebra, then $\mu$ is $f$-invariant?
Try to define $\mathcal{C}_{\epsilon} = \{ C : \| \mu(A) - \mu (f^{-1}(A)) \| < \epsilon\}$
Apr
1
comment If $\mu$ is $f$-invariant for a collection generating the Borel $\sigma$-algebra, then $\mu$ is $f$-invariant?
The family $\mathcal{C}$ is clearly a $\lambda$-system. You could try the $\pi - \lambda$-theorem.
Apr
1
comment If $\mu$ is $f$-invariant for a collection generating the Borel $\sigma$-algebra, then $\mu$ is $f$-invariant?
I guess this is general, but if $\mu$ is a finite measure you can play with the fact that $f^{-1}$ behaves very well under sets operations. From that you can have that $\mathcal{C}$ is in fact an algebra. Maybe there is a way to avoid the sets with infinity measure...
Mar
25
comment Integrals of indicator functions question
Do you look for an upper bound or the equality?
Mar
25
comment Integrals of indicator functions question
What is the relation between $g(x)$ and $g(x,y)$? Could be a typo?
Mar
24
answered Phase trasition of $f(x)$ on random graph $G(n,p(n))$
Mar
24
revised independent random variables probability and measure theory
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Mar
24
suggested approved edit on independent random variables probability and measure theory
Mar
23
answered Is the geometric series of a set of $n$ RVs a martingale?
Mar
23
comment Is the geometric series of a set of $n$ RVs a martingale?
Sum of independent random variables is a martingale if, and only if, $EX_k = 0$.
Mar
17
answered Find $\lim \sup A_n$ and $\lim \inf A_n$?
Mar
17
comment Find $\lim \sup A_n$ and $\lim \inf A_n$?
The definition of $\liminf A_n$ can be interpreted as the set of the $w \in \mathbb{R}^2$ such that exist a $n$ such that $w \in A_m$ to all $m \ge n$.
Mar
12
comment Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale
Yes, I guess you are right. Thanks!
Mar
12
comment Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale
Sorry, but I really didn't understand what you said. Question: A random variable which follows a Cauchy distribution fits the Siryaev's definition? Seems to me that the answer is "yes". And it is natural to define $E[X|X] = X$...
Mar
12
comment Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale
In Shiryaev's book, page 213 the definition doesn't demand integrability. He defines for a positive random variable, doesn't need to be integrable, and then extends to any random variable when $min\{E[X^+| \mathcal{F}],E[X^-| \mathcal{F}]\}< \infty$ almost sure.
Mar
12
comment Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale
@Did, Why do not exists? $E[X | X] = X$ once $X$ is trivially measurable with respect the $\sigma$ algebra generated by $X$. Am I missing something?