$L^1$ distance between Wald-equation like sums of random variables Suppose we have an i.i.d. sequence of random variables $(X_n)_n$ and two dependent random variables $M,N$ which are independent of $(X_n)_n$. Suppose we have a closed expression for $\mathbb{E}[|M-N|]$. Is there an easy way express the distance:
$$
\mathbb{E}\left[\left| \sum_{m=1}^M X_m - \sum_{m=1}^N X_m \right|\right]
$$ 
in function of $\mathbb{E}[|M-N|]$ or find some upper bound for this distance i.f.o. the distance between $M$ and $N$?
 A: The random variable $|\sum_{m=1}^M X_m - \sum_{m=1}^N X_m|$ has the same distribution as $|\sum_{m=1}^{|M-N|} X_m|$, and so: 
\begin{align}
E\left[\left|\sum_{m=1}^M X_m - \sum_{m=1}^N X_m\right|\right] &= E\left[\left|\sum_{m=1}^{|M-N|} X_m\right|\right]\\
&\leq E\left[\sum_{m=1}^{|M-N|}|X_m|\right]\\
&= E\left[E\left[\sum_{m=1}^{|M-N|}|X_m| \: \left|\right. \:  |M-N|\right]\right] \\
&= E\left[\: |M-N| \: E[|X_1|] \: \right] \\
&= E[|M-N|] \: E[|X_1|]
\end{align}

For your question with $X_n$ and $Y_n$ you could use: 
$$ \left|\sum_{m=1}^M X_m - \sum_{n=1}^N Y_n\right| \leq \left|\sum_{m=1}^{\min[M,N]}(X_m-Y_m)\right| + \sum_{m=\min[M,N]+1}^{\min[M,N]+|M-N|} [|X_m|+|Y_m|] $$

For measuring differences in random variables, in addition to the metrics Clement mentions, you could look at the differences in the moment generating functions.  Define $g(r) = E[e^{rX_1}]$ (for $r>0$) and note that (assuming $\{X_i\}$ and $\{Y_i\}$ are i.i.d. with the same distributions, and both independent of $M$ and $N$): 
\begin{align}
E[e^{r\sum_{n=1}^N X_n}] &= E[E[ g(r)^N|N] ] \\
E[e^{r\sum_{m=1}^M Y_m}] &= E[E[ g(r)^M|M] ] 
\end{align}
and  $g(r)^N = g(r)^M g(r)^{N-M}$.
