Convergence in Lp ($X_n\to X\text{ in }{L}^p$)

For $0<p<\infty$, convergence in Lp of real random variables on one probability space means $\mathbb{E}|X_n-X{|}^p\to 0$. For $p\ge 1$ this is convergence in the Lp norm; when $0<p<1$, the same moment criterion is used without claiming it is a norm. The Markov inequality gives $\mathbb{P}(|X_n-X|>\epsilon )\le {\epsilon }^{-p}\mathbb{E}|X_n-X{|}^p$, so it implies convergence in probability. The converse fails: for uniform $U$ on $[0,1]$, $X_n=n^{1/p}{1}_{\{U\le 1/n\}}$ tends to zero in probability but $\mathbb{E}|X_n{|}^p=1$.

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

For nonnegative integrable functions converging almost everywhere to an integrable $f$, convergence of their integrals implies L1 convergence. Indeed $\min (f_n,f)\to f$ almost everywhere and is dominated by $f$, so the dominated convergence theorem gives convergence of its integral. Use $|f_n-f|=f_n+f-2\min (f_n,f)$. This applies to probability densities, giving convergence in total variation distance.

Convergence in $L^2$ means $\mathbb{E}[(X_n-X{)}^2]\to 0$. It implies convergence in probability by Markov inequality.

If square-integrable random variables satisfy $X_n\to X$ almost surely, then $X_n\to X$ in $L^2$ exactly when $\mathbb{E}[X_n^2]\to \mathbb{E}[X^2]$. Boundedness in $L^2$ and almost-sure convergence first give weak convergence in the Hilbert space $L^2$; weak convergence together with convergence of norms gives strong convergence.