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There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. It is obvious that convergence in TV norm implies convergence in distribution. But is there anything we can say about relations between convergence in total variation norm and convergence a.s.,in $L^p$ or in measure?- I did not get any direct hints by googling this, so maybe there is nothing to say about.

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Total variation norm is a norm on measures whereas all the other types of convergence/norms/topologies that you mentioned are on spaces of functions.

Nonetheless functions can be measures too. If you have a sequence of measures $\mu_n$ and say all of them are absolutely continuous with respect to a third measure $\lambda$, then the measures converge in TV norm iff their radon-nikodym derivadives w.r.t. $\lambda$ converge in $L^1$.

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  • $\begingroup$ When those "functions" have as their domain a probability space $\Omega$, i.e. a non-negative-measure space whose total measure is $1$, then those "function" are random variables. Each random variable $X$ has a probability distribution, which is a measure on Borel subsets of $\mathbb R$ (if the random variables are real valued) for which the measure of $(-\infty,a]$ is the measure of $\{\omega\in\Omega : X(\omega)\le a\}$. In that case a sequence of random variables converges in TV distance to a random variable iff the sequence of measures on the line converges in TV distance. $\endgroup$ – Michael Hardy Jul 8 '14 at 0:14

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