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?



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.

  • $\begingroup$ thanks for the resource. Do you know if, in addition to counter examples, it also has the explicit conditions under which you can move between the modes of convergence? I understand one might be able to extract this from careful examination of where the counterexamples, Im just trying to avoid doing all the work if its already been done. Thanks! $\endgroup$
    – Diego
    Dec 2 '15 at 21:44
  • $\begingroup$ What do you mean, move between the modes of convergence? The diagrams in the book, two of which are visible in my link, have arrows. I'll edit my answer to make this clearer. $\endgroup$
    – Unit
    Dec 2 '15 at 21:59

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