Intereting Posts

Manifold notes in more informal way
Congruence question with divisibility
Integrate $ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi$
Is every noninvertible matrix a zero divisor?
Why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$
When does a SES of vector bundles split?
Miller-Rabin-Test : Why can we be $100\%$ certain after $4$ tests for $p=13$?
Integral of the ratio of two exponential sums
Treatise on non-elementary integrable functions
Is $SO_n({\mathbb R})$ a divisible group?
A discrete topological space is a space where all singletons are open $\implies$ all sets are clopen? Closed?
Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
How to compute rational or integer points on elliptic curves
Normal subgroups of the symmetric group $S_N$
Example for an ideal which is not flat (and explicit witness for this fact)

By definition, the weak law states that for a specified large $n$, the average is likely to be near $\mu$. Thus, it leaves open the possibility that $|\bar{X_n}-\mu| \gt \eta$ happens an infinite number of times, although at infrequent intervals.

The strong law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have that for any $\eta > 0$ the inequality $|\bar{X_n}-\mu| \lt \eta$ holds for all large enough $n$.

Now my question is application of these laws. How do I know which distribution satisfies the strong law vs the weak law. For example, consider a distribution $X_n$ be iid with finite variances and zero means. Does the mean $\frac{\sum_{k=1}^{n} X_k}{n}$ converge to $0$ almost surely (strong law of large numbers) or only in probability (weak law of large numbers)?

- Calculate the series expansion at $x=0$ of the integral $\int \frac{xy\arctan(xy)}{1-xy}dx$
- What is the difference between the limit of a sequence and a limit point of a set?
- Is Cesaro convergence still weaker in measure?
- Product of two power series
- Lebesgue Dominated Convergence example
- Find the limit of a recursive sequence

- Does an absolutely integrable function tend to $0$ as its argument tends to infinity?
- Integral of odd function doesn't converge?
- Stone-Čech compactifications and limits of sequences
- Exchange integral and conditional expectation
- measurability of a function -equivalent conditions
- Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?
- Radon–Nikodym derivative and “normal” derivative
- Is the integral $\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$ convergent?
- Convergence in metric and a.e
- prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

If $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with finite mean $\mu$ (in your example, $\mu = 0$),

then by the strong law of large numbers, $\frac{{\sum\nolimits_{i = 1}^n {X_i } }}{n}$ converges to $\mu$ almost surely. In particular, $\frac{{\sum\nolimits_{i = 1}^n {X_i } }}{n}$ converges to $\mu$ in probability. So, you actually don’t have to assume finite variance.

From section 7.4 of Grimmett and Stirzaker’s *Probability and Random Processes (3rd edition)*.

The independent and identically distributed sequence $(X_n)$, with common distribution function $F$, satisfies $${1\over n} \sum_{i=1}^n X_i\to \mu$$ in probability for some constant $\mu$ if and only if the characteristic function $\phi$ of $X_n$ is differentiable at $t=0$ and $\phi^\prime(0)=i \mu$.

For instance, the weak law holds but the strong law fails for $\mu=0$ and symmetric random variables with $1-F(x)\sim 1/(x\log(x))$ as $x\to\infty$.

- Calculate the integral $\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt$, by deformation theorem.
- Why don't I get $e$ when I solve $\lim_{n\to \infty}(1 + \frac{1}{n})^n$?
- Equation of a curve
- Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.
- Positive Elements of $\mathbb{C}G$: as functionals versus as elements of the C*-algebra
- Are Elementary Algebra and Boolean Algebra Algebras over a Ring (or Field)?
- Solving the system $(18xy^2+x^3, 27x^2y+54y^3)=(12, 38)$
- Given $BA$, find $AB$.
- Coupon collector without replacement
- Minkowski's inequality
- Is there any kind of formula to estimate this $1^1+2^2+3^3+…+n^n$?
- What can we say about a locally compact Hausdorff space whose every open subset is sigma compact?
- Dimensions of symmetric and skew-symmetric matrices
- Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$
- What's the arc length of an implicit function?