Intereting Posts

Recommendations for Commutative Algebra Software?
Inverse of State-space representation (control)
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
Examples of Separable Spaces that are not Second-Countable
Proof that the irrational numbers are uncountable
Convolution with Gaussian, without dstributioni theory, part 3
Stopping time proof
$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)
The injectivity of torus in the category of abelian Lie groups
Pullback of a covering map
How to prove that $\sqrt 3$ is an irrational number?
Does $X$ have countable network if it has countable extent?
Lebesgue outer measure of $\cap\mathbb{Q}$
Partitioning a natural number $n$ in order to get the maximum product sequence of its addends
Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

The Central Limit Theorem tells us that for an iid sequence of random variables $(X_n)_{n\geq 0}$ of finite variance $\sigma^2$ and zero mean

$$\lim_{n\to\infty}\frac{S_n}{\sqrt{n}}=^d N(0,\sigma^2)$$

where $S_n=X_1+\cdots+X_n$.

- Understanding the measurability of conditional expectations
- Probability of picking an odd number from the set of naturals?
- Brownian Bridge equivalence of definitions
- Expected value, I do not get this “wikipedia triviality”
- Formula similar to $EX=\sum\limits_{i=1}^{\infty}P\left(X\geq i\right)$ to compute $E(X^n)$?
- Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Suppose we have a similar sequence, except now we suppose that $X_n$ has infinite variance. Then is it possible for the sequence $\frac{S_n}{\sqrt{n}}$ to converge in distribution? Is there always some $\alpha$ such that $n^\alpha S_n$ converges to a non-constant distribution?

(It seems to me that the answer to the first question should be no, but I’m having trouble showing this.)

Thank you.

- Solution to General Linear SDE
- Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}=0$.
- Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion
- Determining if something is a characteristic function
- Intuitive explanation of the tower property of conditional expectation
- Convergence in distribution of conditional expectations
- Is this a Delta Function? (and Delta as limit of Gaussian?)
- Mutual Independence Definition Clarification
- Two-valued measure is a Dirac measure
- Can a time-shifted Brownian motion be also a Brownian motion

The generalized central limit theorem states (see this for a summary), that if $X_i$ are i.i.d. such that its density function has left tail power-law asymptotic $\mathbb{P}(X < -x) \sim d x^{-\mu}$ and right tail asymptotic $\mathbb{P}(X > x) \sim 1- c x^{-\mu}$ as $x \to +\infty$, then there exist sequences $a_n$ and $b_n$ such that the random variate $Z_n = ((\sum_{i=1}^n X_i) – a_n )/b_n$ converges in probability to a stable distribution with stability index $\alpha = \min(\mu, 2)$ and asymmetry parameter $\beta = \frac{c-d}{c+d}$.

Details on the constructive choice of sequences $a_n$ and $b_n$ are given in the table found at the link above. Also see page 62 of Zolotarev and Uchaikin on Google books.

- Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$
- What is the average number of draws (2 cards per draw with shuffles in between) before I had seen all 52 cards in the deck?
- Do non-square matrices have eigenvalues?
- Proving $\mathbb{Q} = \{f(\sqrt{2}): f(x) \in \mathbb{Q}\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$
- Prerequisite reading before studying the Collatz $3x+1$ Problem
- Show the set of points $(t^3, t^4, t^5)$ is closed in $\mathbb A^{3}$
- limit $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$
- Definition of $d (P (x ,y )dx)$
- Proof that $Γ'(1) = -γ$?
- Why is it important to study the eigenvalues of the Laplacian?
- Finding all integer solutions of $5^x+7^y=2^z$
- $\pi(x)\geqslant\frac{\log x}{2\log2}$ for all $x\geqslant2.$
- Limit-Fundamental Concept?
- Why is cardinality of the field important for Noetherian normalization?
- Is the root of $x=\cos(x)$ a transcendental number?