Complete convergence implication diagram

The proved arrows are $L^q(q>p)⟹L^p⟹\text{in probability}⟹\text{in distribution},$ and almost-sure convergence also implies convergence in probability. The only reverse implication here is distributional convergence to a constant.

None of the displayed implications reverses in general. On $(0,1)$, shrinking spikes $(n+1{)}^{1\mathrm{/}p}{\mathbf{1}}_{(0,1\mathrm{/}(n+1))}$ converge almost surely and in probability but not in $L^p$, while the dyadic typewriter sequence converges in every finite $L^p$ but not almost surely. If $X$ is symmetric on $\{-1,1\}$, the constant sequence $X_n=-X$ has the law of $X$ but does not converge to $X$ in probability. Independent indicators with probabilities $1\mathrm{/}(n+1)$ converge in probability but, by Borel--Cantelli, not almost surely. Finally, $(n+1{)}^{1\mathrm{/}q}{\mathbf{1}}_{(0,1\mathrm{/}(n+1))}$ converges in $L^p$ but not $L^q$ when $p<q$. The shrinking spikes $(n+1){\mathbf{1}}_{(0,1\mathrm{/}(n+1))}$ also converge almost surely while their expectations remain equal to one.