Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity. Convergence is also a property of an iterative algorithm to stabilize on some aim value.
Convergence refers to the investigation of the behavior of certain sample quantities when the sample size approaches infinity. Two important types of convergence are convergence in probability and almost sure convergence.
Convergence in probability
A sequence of random variables $X_1,...,X_n$ converges in probability to a random variable $X$ if
$$\lim_{x \to \infty}P(|X_n-X|\leq\epsilon)=1 $$
for every $\epsilon > 0$. This means that at the limit as $n$ increases to infinity almost all of the probability mass becomes concentrated around $X$ in a small interval. This type of convergence is used in the weak law of large numbers.
Almost sure convergence
Similar to the previous statement, a squence of random variables $X_1,...,X_n$ converges almost surely to a random variable $X$ if
$$P(\lim_{x \to \infty}|X_n-X|< \epsilon)=1$$
for every $\epsilon > 0$. Here, compared to the previous case, the limit is achieved with probability one. Almost sure convergence is used in the strong law of large numbers and it implies convergence in probability (note that convergence in probability does not imply almost sure convergence).
