What is the Conditional law? I am sorry to ask such a question, but I always cant understand what is meant by the notation I am currently reading. In a mathematical paper I witness a formula as such
$\mathcal{L}\left( \sqrt{N}\overset{-}{X}\right) \underset{w}{\rightarrow }%
\mathcal{N}\left( 0,c^{2}\sigma ^{2}\right) $
they mentioned that the $\mathcal{L}$ here stands for conditional law given the i.i.d random variables $X_1,X_2,...,X_n$ and $\underset{w}{\rightarrow }%$ stand for weak convergence.I need some further understanding of what they mean. I know usually weak convergence is convergence in measure. while convergence in distribution is related to random variables. 
Can someone give me any further references or anything that might help understand what this notation stand for?! 
Just let me rephrase the question consider random variables $V_{1},V_{2},V_{3},\cdots ,V_{N}$, and a random variable $\overset{-}{X}%
^{\ast }=V_{1}X_{1}+V_{2}X_{2}+V_{3}X_{3}+\cdots +V_{N}X_{N}$, and a constants $x_{1},x_{2,\cdots ,}x_{N}$
Is the following interpretation correct? 
$\mathcal{L}\left( \sqrt{N}(\overset{-}{X}^{\ast }-\overset{-}{X})\right) =%
\mathbb{P}\left( \sqrt{N}(\overset{-}{X}^{\ast }-\overset{-}{X}%
)<x|X_{1}=x_{1},X_{2}=x_{2},\cdots ,X_{N}=x_{N}\right) $
$\mathcal{L}$ here stands for conditional law given the i.i.d random variables $X_1,X_2,...,X_n$
If so what is meant  by $\mathcal{L}$? Is it a random variable or a conditional probability distribution? 
 A: I would take "weak convergence" in this context to mean convergence in distribution.  "Conditional law given" something is a phrase I would take to mean a conditional probability distribution.  But the notation $\mathcal L\left(\sqrt{N}\, \bar X\right)$ looks more like something intended simply to mean the probability distribution of the random variable $\bar X=(X_1+\cdots+X_N)/N$, rather than a conditional distribution.  Elsewhere in your question you use the lower-case $n$.  Your question seems less than comprehensible unless you intend that to mean the same thing as the capital $N$, and you shouldn't be using the capital and lower-case letters interchangeably.
Congergence in distribution to $\mathcal N(0,c^2\sigma^2)$ would mean that the cumulative distribution function of $\sqrt{N}\,\bar X$ converges pointwise to the cumulative distribution function of $\mathcal N(0,c^2\sigma^2)$, except at points where $\mathcal N(0,c^2\sigma^2)$ is discontinuous (and there are no points where $\mathcal N(0,c^2\sigma^2)$ is discontinuous, so one need not consider the exception in this case).
