Let $X_n = 1,2,\dots$ be a sequence of random variables. Then, convergence of the random variables in

Quadratic Mean (or Mean Square). $X_n \xrightarrow{ q.m. } X$

$$\lim_{n\rightarrow \infty} \mathsf{E}[(X_n – X)^2]=0$$

the strong convergence implies the below $\mathbf{\Downarrow}$

Probability. $X_n \xrightarrow{ P } X$

$$\lim_{n\rightarrow \infty} \mathsf{P}(|X_n – X| \geq \varepsilon )=0,$$
for $\varepsilon >0$. That is,
$$\plim_{n\rightarrow \infty} X_n = X$$

which in turn implies $\mathbf{\Downarrow}$

Distribution(or convergence in law). $X_n \xrightarrow{ d } X$

$$\lim_{n\rightarrow \infty} F_n (x) = F(x)$$which has the weakest convergence. For all $x$ at which the cumulative distribution function (CDF) $F(x) = \mathsf{P} (X\leq x)$ of $x$ is continuous.

The strongest convergence (not included above) is the Almost Sure $X_n \xrightarrow{ a.s } X$ convergence.