TITLE: Convergence in probability and Dirac Delta function
QUESTION [1 upvotes]: When speaking about convergence in probability isn't the limit of the difference of two random variables actually the Dirac Delta Function? So is in a sense, the Dirac Delta function playing the role of a zero in the space of random variables?
Thanks in advance!

REPLY [1 votes]: It seems like you are actually thinking about convergence in distribution, rather than convergence in probability. 
Convergence in distribution of random variables is related to weak convergence of probability measures. More specifically, $X_n\to X$ in distribution if and only if $\mu_n\to\mu$ weakly, where $\mu_n$ and $\mu$ are the distributions of $X_n$ and $X$.
The distribution of the random variable which is 0 for all outcomes is the point mass $\delta_0$. Thus, $X_n\to 0$ in distribution if and only if $\mu_n\to\delta_0$ weakly.
Now, convergence in probability implies convergence in distribution. So if $X_n\to0$ in probability, then $\mu_n\to\delta_0$ weakly (but the converse is not true).
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