Rodrigo Ribeiro
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 Apr13 revised Could we define two random variables such that the product of them is Normal distribution(Gaussian)? improved formatting Apr13 suggested approved edit on Could we define two random variables such that the product of them is Normal distribution(Gaussian)? Apr3 revised Conditional Probability in Geometric Distribution improved formatting Apr3 suggested approved edit on Conditional Probability in Geometric Distribution Apr1 comment If $\mu$ is $f$-invariant for a collection generating the Borel $\sigma$-algebra, then $\mu$ is $f$-invariant? Try to define $\mathcal{C}_{\epsilon} = \{ C : \| \mu(A) - \mu (f^{-1}(A)) \| < \epsilon\}$ Apr1 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. Apr1 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... Mar25 comment Integrals of indicator functions question Do you look for an upper bound or the equality? Mar25 comment Integrals of indicator functions question What is the relation between $g(x)$ and $g(x,y)$? Could be a typo? Mar24 answered Phase trasition of $f(x)$ on random graph $G(n,p(n))$ Mar24 revised independent random variables probability and measure theory Improved formatting Mar24 suggested approved edit on independent random variables probability and measure theory Mar23 answered Is the geometric series of a set of $n$ RVs a martingale? Mar23 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, $EX_k = 0$. Mar17 answered Find $\lim \sup A_n$ and $\lim \inf A_n$? Mar17 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 $m \ge n$. Mar12 comment Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale Yes, I guess you are right. Thanks! Mar12 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 $E[X|X] = X$... Mar12 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. Mar12 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?