A more "complete picture" of the relationship between various modes of convergence

Question

I am trying to generate/looking for a more comprehensive/complete list/diagram of how the 4 major modes (as listed on wikipedia) of convergence of random variables relate to each other:

• Distribution (law)
• Probability
• Almost sure
• $\mathcal{L}^p$ (in mean)

Wikipedia also has this handy little chart (image versions of this exist online as well) $$\begin{matrix} \xrightarrow{L^s} & \underset{s>r\geq1}{\Rightarrow} & \xrightarrow{L^r} & & \\ & & \Downarrow & & \\ \xrightarrow{a.s.} & \Rightarrow & \xrightarrow{\ p\ } & \Rightarrow & \xrightarrow{\ d\ } \end{matrix}$$

What I am trying to generate/looking for is something like this: $$ \begin{matrix} \xrightarrow{L^s} & \underset{s>r\geq1}{\Rightarrow} & \xrightarrow{L^r} & & \\ & & \Downarrow \overset{(a)}{\uparrow} & & \\ \xrightarrow{a.s.} & \underset{\overset{(b)}{\leftarrow}}{\Rightarrow} & \xrightarrow{\ p\ } & \Rightarrow & \xrightarrow{\ d\ } \end{matrix}$$

Where: $(a) = $ "with uniform integrability" and $(b) = \text{ if } \forall \epsilon> 0, \sum_n \mathbb{P} \left(|X_n - X| > \varepsilon\right) < \infty$

and so on and so forth. Does anyone know if anything like this exists? Does anyone want to help me fill this out? Can anyone suggest a better format for organizing this?

Thanks!

Answer 1

Gearoid de Barra's 2003 book Measure Theory and Integration contains an entire section called "Convergence Diagrams" (7.3, pp.128-131) concerning six modes of convergence in three different settings: almost everywhere, in mean, uniform, in $L^p$, almost uniform, and in measure. The settings are: the general case, the case where $\mu(X) < \infty$, and the case where the sequence of converging functions is dominated by an $L^1$ function. In each setting, the diagram features arrows between modes of convergence: an arrow from mode 1 to mode 2 means, in that setting, convergence in mode 1 implies convergence in mode 2. For example, in the general case, almost uniform convergence implies almost everywhere convergence; if the space has finite measure, the converse is true, and the arrow is there to show this.

The subsequent section devoted to counterexamples illustrating the non-implications between the modes, which explain why the diagrams lack arrows where they do. Some of the relevant pages are visible on Google Books.