# Relations among notions of convergence

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Does this imply that $plim_{n\rightarrow \infty}A_n=0$, where $plim$ is the probability limit?

• Are $A_n$ supposed to be random variables? What is a probability limit for real numbers? Oct 30 '14 at 17:53
• $A_n$ are real numbers.
– TEX
Oct 30 '14 at 17:57
• A sequence of real numbers is a sequence of random numbers with trivial distribution. Oct 30 '14 at 18:31
• 1) It is not a good idea ask another question in comments. 2) Where have you find the notation $plim$ probability limits? Oct 30 '14 at 18:50
• in the Econometric book Hayashi
– TEX
Oct 30 '14 at 18:53

$$P( |A_n-0|> \epsilon) \leq \frac{E(|A_n|)}{\epsilon}=\frac{|A_n|}{\epsilon}$$ Letting $n\rightarrow \infty$ on both sides proves convergence in probability.
$X_n\to X$ in probability iff $\forall \epsilon>0$ $\lim\limits_{n\to\infty}P\{|X_n-X|>\epsilon\} = 0$.
We should concern sequence of random numbers $X_n$ such that $P\{X_n=A_n\}=1$. As $\lim\limits_{n\to\infty}A_n=0$ then for any $\epsilon>0$ there is $n_\epsilon$ such that $|A_n|<\epsilon$ for any $n>n_\epsilon$ and the same for $X_n$. It implies that $P\{|X_n-0|>\epsilon\}=0$ for $n>n_\epsilon$.
So we have that for any $\epsilon>0$ $\lim\limits_{n\to\infty}P\{|X_n-0|>\epsilon\} = 0$.