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Results tagged with probability-theory
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user 9849
For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.
0
votes
Accepted
Prove that the given function is in $L^1$
Hint:
$$\left|g(\omega)+\sup_{k\geqslant M}|g_k(\omega)-g(\omega)|\right|\leqslant 2|g(\omega)|+\sup_{k\geqslant 1}|g_k(\omega)|\leqslant 3\sup_{k\geqslant 1}|g_k(\omega)|.$$
Indeed, we have for any …
- 175k
2
votes
Accepted
tightness under a certain uniform integrability condition
(2) implies tightness because we have $X_k\chi_{|X_k|\gt L}\geqslant L\chi_{|X_k|\gt L}$, which gives
$$\lim_{L\to \infty}L\sup_n\mu_n\left(|X_k|\gt L\right)=0.$$ So for a fixed $\varepsilon$, take $L …
- 175k
0
votes
Accepted
definition of expected value
We can see that the first and second definition are equivalent by Fubini's theorem. Indeed, we can write $X:=X^+-X^-$ where $X^+,X^-$ are non-negative, then we use the fact that for a non-negative ran …
- 175k
0
votes
Expectation of squared random variable
A bounded random variable has a finite expectation.
We have $|W|\leqslant n^2$.
- 175k
4
votes
Accepted
Dropping some hypotheses on a result involving convergence in distribution
Portmanteau states that converges in distribution of $X_n$ to $X$ is equivalent to $\mu(X_n\in A)\to \mu(X\in A)$ for all $A$ such that $\mu(\partial A)=0$, where $\partial A$ is the boundary of $A$. …
- 175k
0
votes
What is the limiting distribution of $\sum_{i=1}^{n}(Z_{i}+1/n)/\sqrt{n}$?
Start from
$$
\sum_{i=1}^{n}\dfrac{Z_{i}+1/n}{\sqrt{n}}=\sum_{i=1}^{n}\dfrac{Z_{i}}{\sqrt{n}}+\frac{1}{\sqrt n}.
$$
Then using the assumption on the random $Z_i$, $\sum_{i=1}^{n}\frac{Z_{i}}{\sqrt{n}} …
- 175k
0
votes
Accepted
Probabilistic statement
Let $l\geq 1$. Since $X_n$ gets its values on $\mathbb N$, the set $\{\omega\in\Omega, X(\omega)=l\}$ is equal the set $\{\omega\in\Omega, \exists N(\omega)\mid X_n(\omega)=l \,\forall n\geq N(\omega) …
- 175k
5
votes
If $E|X|<+\infty$ then $\int_{\left(|X|>n\right)}X \;dP\to 0$
Put $A_k:=\left\{\omega\in\Omega,k\leq |X(\omega)|<k+1\right\}$. We have for all $k\geq 0$:
$$kP(A_k)\leq \int_{k\leq X<k+1}|X|dP\leq (k+1)P(A_k)$$
hence
$$\sum_{k=0}^{+\infty}kP(A_k)\leq E|X|\leq\su …
- 175k
1
vote
Accepted
Convergence of moments and absolute moments of random variables
We have using Hölder's inequality and denoting $M:=\sup_{n\geq 1}E|X_n|^r$
\begin{align*}
\int |X_n|^{\alpha}\mathbf 1_{|X_n|^{\alpha}> m}&\leq \left(\int |X_n|^r\right)^{\alpha/r}\left(P(|X_n|^\alph …
- 175k
2
votes
Accepted
limit superior of uniformly integrable random variables is integrable?
Take $\{X_n\}$ a sequence of i.i.d. , with a density of unbounded support, and integrable.
The supremum in the definition of uniform integrability $\sup_{n\in\Bbb N}\int_{\{|X_n|\geq R}|X_n|dP$ is a …
- 175k
2
votes
Under weak convergence, is the limit's moment infinite if the sequence's moments are infinite?
Let $X_n$ of density $f_n(x):=\frac{2n}{\pi(1+nx^2)}\chi_{(0,+\infty)}$; if $g$ is a continuous bounded function then
$$\int_0^{+\infty}(f_n(x)g(x)-f_n(x)g(0))dx\leq 2\int_0^{+\infty}\frac{|g(tn^{-1} …
- 175k
1
vote
Accepted
generated $\sigma$-algebras
The map $T\colon(t_1,\ldots,t_n)\mapsto (t_1-t_2,t_2-t_3,\ldots,t_{n-1}-t_n,t_n)$ is linear and bijective. The set
$$\{X_1-X_2\leq x_1',X_2-X_3\leq x_2',\ldots,X_{n-1}-X_n\leq x'_{n-1},X_n\leq x_n'\} …
- 175k
3
votes
Equivalence of probability measures
$P(A)=0$ doesn't imply that $\Bbb 1_A\equiv 0$ (but it implies that this indicator function is in the class of $0$ for the equivalence $P$-almost everywhere, and it's probably what you meant). Since y …
- 175k
1
vote
Accepted
Showing convergence in distribution example
Write $$U_n:=\color{green}{\frac{X_1+\cdots+ X_n}{\sqrt n}}\cdot\color{red}{\frac n{X_1^2+\dots+X_n^2}},$$
then use and show the following:
If $Y_n\to Y$ in distribution and $Z_n\to c\neq 0$ in pr …
- 175k
2
votes
Accepted
Formalizing the shift operator
Actually, $T_A$ act on the sequence $\{X_n(\omega)\}_{n\geq 0}$. Hence we have for a fixed $\omega$:
$$T_A(\theta_n\{X_k(\omega)\})=T_A(\{X_{n+k}(\omega)\})=\inf\{j>0,X_{n+j}\in A\}=\inf\{k>n,X_k\in A …
- 175k