Questions tagged [convergence-probability]
11 questions
3
votes
2 answers
probability convergence as compactness for a.s. convergence
I stumbled upon the sequential characterization of probability convergence.
$ X_n \xrightarrow{\mathbb{P}} X$ if and only if for each subsequence $X_{n_k}$, there exists a subsequence $X_{n_{k_j}}$ such that $X_{n_{k_j}} \xrightarrow{a.s.}…
Simone Licciardi
- 394
2
votes
1 answer
Stochastic convergence with and without rate
I have a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$. If I know that there exists a sequence of strictly positive real numbers $(\epsilon_n)_{n \in \mathbb{N}}$ which is such that $\lim_{n \rightarrow \infty} \epsilon_n = 0$…
Akurishen
- 83
1
vote
0 answers
Proof check/help for $\xrightarrow{\text{a.s.}}$ $\Rightarrow$ $\xrightarrow{\text{P}}$
I tried to show that $X_n \xrightarrow{\text{a.s.}} X \Rightarrow X_n \xrightarrow{\text{P}} X$, but my proof is little different from ones I've seen because it works with the limit definition. I feel uneasy about a step in the middle involving…
rudinable
- 229
1
vote
1 answer
How to get the Corollary from the Theorem
I would like to understand how this Theorem implies the subsequent Corollary:
Theorem. Let A be an irreducible matrix, related to a continuous time Markov branching process $X(t)$, with dominant eigenvalue $\lambda_1$ and associated eigenvector…
Dada
- 177
- 6
- 22
1
vote
2 answers
Assume $X_n y \to 0$ in probability for all $y \in S$, and let $Y$ be a $S$-valued RV, independent of $X$. Show that $X_nY \to 0$ in probability.
I came across the following probability question. Let $(X_n)_{n=1}^{\infty}$ be a sequence of random variables, and let $S \subset \mathbb{R}$ be a measurable set. Assume that for each $y \in S$, we have that:
$$X_ny \xrightarrow[n \to \infty]{} 0…
mathematico
- 463
1
vote
0 answers
$O_P$ and $o_P$ calculation
Let's say $A_n=O_p(n^{-3/5}\sqrt{log n})$. I wonder whether I can express $A_n$ as $o_p(n^{-2/5})$.Here's my attempt.
$$ P(\frac{|A_n|}{|n^{-3/5}\sqrt{log n}|} > \lambda) < \varepsilon $$ for all $n > M$ for some $M >0$. This implies that
$$P(|A_n|…
Lee
- 163
1
vote
1 answer
MIT Statistic For Applications course Question 1
I'm having trouble understanding the answer to one of the mit opencourseware Statistic For Applications homeworks
The question: (question 1.): [0]
The answer: [1]
The question asks to show this random variable converges in probability:
$P[X_n=1/n] =…
tom tom
- 29
1
vote
1 answer
Convergence of positive random vector
Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$.
That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$.
Suppose I further have that $\mathbb{E}\left[ \sum_{i=1}^l \left(…
1
vote
0 answers
Problem on convergence in distribution of a random vector
The issue is that I have to prove the following with limited resources.
Problem: Let $X_n$ and $Y_n$ be p-dimensional random vectors. Show that if $X_n − Y_n \xrightarrow{P} 0$ and $X_n \xrightarrow{D} X$ , where $X$ is a p-dimensional random…
0
votes
2 answers
show that if $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$
I have the following question:
Show that if $(Y_n)$ be a sequence of random variables that satisfies $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$.
I've proceeded as follows but I'm not sure if I'm formally…
ZedVeZed
- 51
0
votes
1 answer
limit of convergane in probability
I was learning about the convergence in probability. I'm unsure if looking at the epsiolon-delta definitions below which captures better the convergence in probability correctly.
$$ \forall \epsilon,\delta>0 \quad \exists n>N \qquad S.T \qquad…
