Lotus BlogTheory - [T11]
Mathematical Convergence
Convergence in mathematics is the property, displayed by functions or infinite numbers series, of approaching a limit (specific value), as an argument increases or decreases more and more. To clarify we can take the function represented by $ f(x) = \frac{1}{x} $ if we now express the limit represented by the value of $x$ (the argument) reaching towards infinite ($\infin$) we will see that the function will get closer and closer to the value of $0$.
$$ \lim_{x\rightarrow\infin} \frac{1}{x} = 0 $$
Convergence in probability
Convergence in probability is a specific type of convergence that relates to Random Variables. In particular it provides informations related to the likelihood of a RV approaching a constant value as a Sample Set grows larger and larger. This can be expressed by the following notation:
$$ \lim_{n\rightarrow\infin} P(|X_n - \mu| > \epsilon) = 0 $$
To clarify the given formula, we can see $\mu$ as the constant value to which the RV $X$ converges as $n$ grows towards infinity. Taken $\epsilon$ to be a small value we can observe that the probability that the “distance” between the RV $X$ and the constant $\mu$ is greater than the value of $\epsilon$ goes towards $0$. Meaning that the RV $X$ will convergence in probability towards the value of $\mu$. The application developed in the last homework is a great visual example of convergence in probability.