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 6m comment Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$ Before jumping into rather mysterious considerations such as $\sup X_k>1/N$ (for integer valued random variables, what the...), you might want to think more about what the event $[X_n>0\ \text{i.o.}]$ really means. 9m comment Proving that $\log x$ is Big Oh of $x^k$ for every positive k One can hypothetize that this answer is not useful to anybody asking the question. In particular, the trick showing that the $k=1$ case implies the case of every $k>0$ is not explained. 38m comment Sufficient conditions for a probability measure to be characterized by finite number of moments Re giving a "hint to solve a differential equation like the one above", note that the first identity in the post is rather mysterious, equating random variables $g(Y(x))$ and $F_y(g(Y(x)))$ (which should probably read $F_{Y(x)}(g(Y(x)))$...) with numbers $E(Y(x))$ and $\frac{\mathrm d}{\mathrm d x} E(Y(x))$. The first sentence of the post does not compute either (what is "a random variable defined on a probability measure" supposed to mean?). Please explain. 49m comment Is this a vector field? Even the suggested rescaling does not make that the vector field corresponds to the differential equation. 52m comment Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$ Yeah--in other words, you rediscovered the standard proof of the easy part of Borel-Cantelli... 54m comment Write out the explicit Kolmogorov forward differential equation Use the decomposition $$P(X_{t+h}=1)=P(X_{t+h}=1\mid X_t=1)P(X_{t}=1)+P(X_{t+h}=1\mid X_t=2)P(X_{t}=2),$$ that is, $$f(t+h)=(1-ah+o(h))f(t)+(bh+o(h))g(t),$$ or equivalently, $$f(t+h)-f(t)=(-af(t)+bg(t))h+o(h),$$ that is, $$f'(t)=-af(t)+bg(t),$$ and now note that $g(t)=1-f(t)$... 1h comment Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$ Hint: Borel-Cantelli lemma (both parts). 1h comment Convolution of $te^{2t}$ and $\delta_1-\delta_2$? Good. Now you know. 1h comment Zeno's arrow paradox @fleablood As always, and as you probably know, "philosophers" are a diverse bunch of people, some are declaring they discovered some fundamental paradigm shift every other morning (and those might be prone to rediscover this specific example and wax on it endlessly) while others are doing more serious stuff and are well aware of what you say. 1h comment Convolution of $te^{2t}$ and $\delta_1-\delta_2$? I don't get it, why not use $$f\ast(\delta_1-\delta_2)=(f\ast\delta_1)-(f\ast\delta_2),$$ and then your (2)? 1h comment Proving that $\log x$ is Big Oh of $x^k$ for every positive k If you know that $\log x=O(x)$, use $\log x=\frac1k\log (x^k)$ to deduce that $\log x=O(x^k)$ for every positive $k$. 1h comment Why is $\frac{1}{1-x} = 1 + \Theta(x)$ for $x \in (0,1)$? Actually, this is wrong, $\frac1{1-x}=1+f(x)$ with $f(x)\gg x$ when $x\to1$. 1h comment Inversion of the inequality sign when raising to a negative power Because $x\mapsto x^{-1/2}$ is decreasing on its domain $x>0$. Or, because $x\mapsto 1/x$ is decreasing on $x>0$ and $x\mapsto x^{1/2}$ is increasing on its domain $x>0$. On the other hand, $\exp$ is increasing hence $a>b$ implies $e^a>e^b$ for every $a$ and $b$ real numbers. 3h comment Prove that $N_n(f+g)=N_n(f)+N_n(g)$ Hint: with disjoint support. 3h comment Zeno's arrow paradox "I also asked these question in there" Yeah, 1 hour ago and 12 hours ago, and you have no answer yet. What is this world coming to... 3h comment Convergence of $\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}$ The change of variable $x=e^t$ and the inequality $e^t>t$ yield $$\int\frac{dx}{\ln(x)^2}=\int\frac{e^tdt}{t^2}>\int\frac{dt}t,$$ from which one can conclude. 3h comment why mixed normal distribution has heavier tail than normal distribution? Any source for this (dubious) claim? 14h comment Questions on probability law What is (6), already? 15h comment How can I calculate definite integral of chi-squared pdf with one degree of freedom Everything works... Try the change of variable $x=t^2$ and check the normal PDF. 15h comment Dynamical Systems: Disease model, what happens to variable $m$? Equations (1)-(2)-(3) describe a restricted version of the model, when $m=0$. This might be explained on the previous page.