Intereting Posts

Continuous functions and uncountable intersections with the x-axis
Find pairs of side integers for a given hypothenuse number so it is Pythagorean Triple
$3\times3$ linear system organization
Showing a Ring of endomorphisms is isomorphic to a Ring
What was Lame's proof?
Minimal sets of generators for groups
Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$
How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification
Why is the derivative of a vector orthogonal to the vector itself?
Quotient of a Clifford algebra by its radical is a Clifford algebra?
To show that Fermat number $F_{5}$ is divisible by $641$.
Equivalent form of prime number theorem
Convergence of $\sum^\infty_{n=1} \ln(1+\frac 1 {2^n})$
Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?
What is $\int x! $ $ dx$?

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$

for every $r\ge 1$

This is the very notation used. I believe it should be:

$$E[|X_n|^r]\rightarrow E[|X|]^r $$

**Attempt** I think I can obtain $E[X_n]\rightarrow E[X]$ using Jensen Inequality but I don’t think this helps. I have no further idea.

- Two equivalent definitions of a.s. convergence of random variables.
- Dropping the “positive” and “decreasing” conditions in the integral test
- Dirichlet's function expressed as $\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)$
- Convergence in mean
- Determine $x$ such that $\lim\limits_{n\to\infty} \sqrt{1+\sqrt{x+\sqrt{x^2…+\sqrt{x^n}}}} = 2$
- Proving convergence of $\sum \frac{\sin n}{2^n}$

- modeling with exponential distributions
- What is the probability of having a pentagon in 6 points
- Improving Von Neumann's Unfair Coin Solution
- Space Sobolev $W^{m,p}$ complete
- The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?
- Probability of rolling a die
- Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i} $
- relative size of most factors of semiprimes, close?
- Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :
- Is statistical dependence transitive?

For any norm $\|\cdot\|$, by the triangle inequality,

$$

\|X_n\|\le\|X_n-X\|+\|X\|

$$

and

$$

\|X\|\le\|X-X_n\|+\|X_n\|

$$

so that

$$

|\|X_n\|-\|X\||\le\|X_n-X\|.

$$

This property is sometimes called the continuity of the norm.

$(\operatorname E|X|^r)^{1/r}$ is the norm of the space of random variables with $\operatorname E|X|^r<\infty$ for $r\ge1$. We obtain that

$$

|(\operatorname E|X_n|^r)^{1/r}-(\operatorname E|X|^r)^{1/r}|\le(\operatorname E|X_n-X|^r)^{1/r}.

$$

- Show that $C_0(, \mathbb{R})$ is not $\sigma$-compact
- Complex numbers equation: $z^4 = -16$
- A group with five elements is Abelian
- How to compute $2^{\text{some huge power}}$
- Does the improper integral $\int_0^\infty e^{-x^2}dx$ converge?
- If $f:\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in $
- Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$
- Motivation behind the definition of tangent vectors
- Given a matrix $A$ with a known Jordan decomposition, what is the Jordan decomposition of $A^2+A+I$?
- Characterisation of inner products preserved by an automorphism
- What are better approximations to $\pi$ as algebraic though irrational number?
- What is a type?
- Finding specific ideals of a ring
- Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?
- How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}$