Definition of Stable convergence in law: why do we need an extension of the probability space?

Question

I am trying to understand the definition of stable convergence in law. I have found the following definition.

Definition. Let $Y_n$be a sequence of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with value in $\mathbb{R}^d$. We say that $Y_n$ converges stably with limit $Y$, where $Y$ is defined on an extension $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$, iff for any bounded, continuous function $g$ and for any bounded $\mathcal{F}$-measurable random variable $Z$ it holds that

$$ \mathbb{E}[g(Y_n)\,Z]\longrightarrow\mathbb{E}^{\prime}[g(Y)\,Z]\text{ as }n\rightarrow\infty $$

My problem is: why do we need an extension of the probability space? And how this extension is generally defined?

Answer 1

Lemma : The following are equivalent $$ Y_n\rightarrow Y \quad stably \\ (Y_n,Z)\xrightarrow{d}(Y,Z) $$ for any $\mathcal{F}$-mb. rv. $Z$.

Proposition: Assume $Y_n\rightarrow Y$ stably, where $Y$ is $\mathcal{F}$-mb. Then $Y_{n}\xrightarrow{P} Y$.

Proof: Since $ Y_n\rightarrow Y \quad stably$ and $Y$ is $\mathcal{F}$-mb, we know by the Lemma, that $(Y_n,Y)\xrightarrow{d}(Y,Y)$. Thus $Y_n -Y\xrightarrow{d} 0$ which means, that $Y_n\xrightarrow{P} Y$.

Stable convergence is a property of the random variables itself, not their distribution functions. Thats why you define its limit on an extension.

However your Definition connects two concepts, the stable convergence, and the weak $L_{1}$-conergence.