3

Consider the following definition.

Definition

Let $X_n$ be a sequence of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with value in $\mathbb{R}^N$. We say that the sequence $X_n$ converges stably in distribution with limit $X$, written $X_n\stackrel{\text{st}}{\longrightarrow} X$, if and only if, for any bounded continuous function $f:\mathbb{R}^N\to\mathbb{R}$ and for any $\mathcal{F}$-measurable random variable $Z$, it happens that: $$ \lim_{n\rightarrow \infty}\mathbb{E}[f(X_n)\,Z]=\mathbb{E}[f(X)\,Z]. $$

I want to prove that if $$ Y_n\stackrel{\text{st}}{\longrightarrow} Y $$ then $$ (Y_n,Z)\stackrel{\text{d}}{\longrightarrow}(Y,Z), $$ for any measurable $Z$ and where $\stackrel{\text{d}}{\longrightarrow}$ indicates the standard convergence in distribution.

My idea is to use the Portmanteau Lemma, so I want to prove that, for any continuous bounded function $g$ it holds that $$ \lim_{n\rightarrow \infty}\mathbb{E}[g(Y_n,Z)]=\mathbb{E}[g(Y,Z)].\quad (1) $$ Since $Y_n\stackrel{\text{st}}{\longrightarrow} X$ I know that for any continuous bounded function $f$ it holds that
$$ \lim_{n\rightarrow \infty}\mathbb{E}[f(Y_n)\,Z]=\mathbb{E}[f(Y)\,Z],\quad (2), $$ however I do not know how to pass from (2) to (1).

===================================================

As suggested below I tried with Levy continuity theorem, but I am not sure that it is ok to consider a complex $f$. This is my poof:

enter image description here

AlmostSureUser
  • 1,180
  • Probably I have found a solution, quite simple indeed. For every $g(x,z)$ bounded and continuous define $$f(x) = g(x,z)/z$$ which is bounded and continuous in $x$. – AlmostSureUser Apr 12 '18 at 14:00
  • 3
    From my point of view, it would be easier to use Lévys continuity theorem, i.e. the fact that convergence in distribution is equivalent to convergence of the characteristic functions. – saz Apr 12 '18 at 14:00
  • @saz would this require the adoption of a complex $f$ ? – AlmostSureUser Apr 12 '18 at 15:06
  • Yes, but for this you can write $f = \text{Re} , f+ i , \text{Im f}$ and use the linearity of the expectation – saz Apr 12 '18 at 15:22

0 Answers0