probability theory - Find a limit $lim_{ctoinfty} Ebig[ X e^{-frac{(X-c)^2}{2}} big] big/ Ebig[e^{-frac{(X-c)^2}{2}}big]$

Let $X$ be a bounded random variable, i.e., $|X| \le a$.
Can we find
$$\begin{align} g(c)=\frac{E\left[ X e^{-\frac{(X-c)^2}{2}} \right]}{E\left[e^{-\frac{(X-c)^2}{2}}\right]} \end{align}$$

How to find the following limit
$$\begin{align} \lim_{ c \to \infty} g(c) \end{align}$$

I tried L'Hopital rule but it did not work out.

Answer

The intuition comes from the special case where $X$ is discrete and takes only finitely many values with probability 1. Indeed, assume that $x_1 < \cdots x_n$ and $p_i := \Bbb{P}(X = x_i) > 0$ for each $i = 1, \cdots, n$ and $p_1 + \cdots + p_n = 1$. Then

$$g(c) = \frac{\sum_{i=1}^{n} x_i p_i e^{-(c-x_i)^2/2}}{\sum_{i=1}^{n} p_i e^{-(c-x_i)^2/2}} \xrightarrow[\ c\to\infty \ ]{} x_n = \operatorname{esssup}X.$$

For general cases, let $\alpha < \beta < \operatorname{esssup}X$ and consider

$$M(c) = \Bbb{E}[e^{-(X-c)^2/2} \mathbf{1}_{\{X > \alpha\}} ] \quad \text{and} \quad m(c) = \Bbb{E}[e^{-(X-c)^2/2} \mathbf{1}_{\{X \leq \alpha\}} ].$$

We assume $c > a$ WLOG so that $x \mapsto e^{-(x-c)^2/2}$ is increasing on $[-a, a]$. Then from the following simple estimates

$$M(c) \geq \Bbb{E}[e^{-(X-c)^2/2} \mathbf{1}_{\{X \geq \beta\}} ] \geq e^{-(\beta-c)^2/2} \Bbb{P}(X \geq \beta) \quad \text{and} \quad m(c) \leq e^{-(\alpha-c)^2/2}$$

we easily find that

$$\frac{m(c)}{M(c)} \leq \frac{e^{(\beta^2-\alpha^2)/2}}{\Bbb{P}(X \geq \beta)} e^{-c(\beta-\alpha)} \xrightarrow[ \ c \to \infty \ ]{} 0.$$

Now by writing

$$g(c) = \frac{\Bbb{E}[X e^{-(X-c)^2/2} \mathbf{1}_{\{X > \alpha\}}] + \Bbb{E}[X e^{-(X-c)^2/2} \mathbf{1}_{\{X \leq \alpha\}}]}{\Bbb{E}[e^{-(X-c)^2/2} \mathbf{1}_{\{X > \alpha\}}] + \Bbb{E}[e^{-(X-c)^2/2} \mathbf{1}_{\{X \leq \alpha\}}]} \geq \frac{\alpha M(c) - a m(c)}{M(c) + m(c)}$$

it follows that

$$\alpha \leq \liminf_{c\to\infty} g(c) \leq \limsup_{c\to\infty} g(c) \leq \operatorname{esssup} X.$$

Taking $\alpha \uparrow \operatorname{esssup}X$ completes the proof.