Intereting Posts

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How to find the minimum of $a+b+\sqrt{a^2+b^2}$
Counting Irreducible Polynomials
$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$
How can I find a subset of a set with “half the size” of the original?
Using Fermat's Little Theorem, find the least positive residue of $3^{999999999}\mod 7$
Inequality involving sums of fractions of products of binomial coefficients
Show that if $ab \equiv ac$ mod $n$ and $d=(a,n)$, then $b \equiv c$ mod $\frac{n}{d}$
How to see $\sin x + \cos x$
$g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$
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Can the golden ratio accurately be expressed in terms of e and $\pi$
What is the expected time to cover $m \leq N$ elements in a set by sampling uniformly and with replacement?
Integer coordinate set of points that is a member of sphere surface
an example of a continuous function whose Fourier series diverges at a dense set of points

This is a follow-up to this question.

Now let $(X_n)$ be a sequence of *positive* random variables. Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=\mu$. Does that imply that $(X_n)$ converges to $\mu$ in mean, i.e., that $\lim_{n\rightarrow\infty}\mathbb{E}[|X_n-\mu|]=0$?

My previous question was for a general sequence of random variables, and Deven Ware showed that the above implication does not hold in the general case…

- Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$?
- Proving a sequence defined by a recurrence relation converges
- Mean of a Convergent Sequence
- Lebesgue Dominated Convergence example
- Convergence of $\sum_n \frac{n!}{n^n}$
- What IS conditional convergence?

- Is there anything special about a transforming a random variable according to its density/mass function?
- Find a sequence of r.v's satisfying the following conditions
- Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$
- Inequality regarding norms and weak-star convergence
- Determine $x$ such that $\lim\limits_{n\to\infty} \sqrt{1+\sqrt{x+\sqrt{x^2…+\sqrt{x^n}}}} = 2$
- Proof of “continuity from above” and “continuity from below” from the axioms of probability
- Convergence of Lebesgue integrals
- $\ell_1$ and unconditional convergence
- Expression for $n$-th moment
- Finding a Distribution When Introducing an Auxiliary Random Variable

Of course not. Try $X_n=X_1$ for every $n$, with $X_1\geqslant0$.

The WP page on the convergence of random variables might help you delineate some plausible implications in this context.

- Is the max of two differentiable functions differentiable?
- Cohen-Macaulay ring and saturated ideal
- Reducibility of $P(X^2)$
- Is there a rational surjection $\Bbb N\to\Bbb Q$?
- Open cover rationals proper subset of R?
- Why are continuous functions not dense in $L^\infty$?
- Is it true that $E = X$?
- $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
- Central limit theorem and convergence in probability from Durrett
- Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
- Plotting a Function of a Complex Variable
- Most even numbers is a sum $a+b+c+d$ where $a^2+b^2+c^2=d^2$
- Prove that any two cyclic groups of the same order are isomorphic?
- Proving that $\cos(2\pi/n)$ is algebraic
- Proving “The sum of n consecutive cubes is equal to the square of the sum of the first n numbers.”