Approximating a random variable by a sequence of random variables

Question

Consider the triangular hat function:

\begin{equation} \varphi(x) = \begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases} \end{equation}

It is well known that $\varphi$ is the pdf $$Y = X_1 + X_2 - 1,$$ where each $X_i$ is a uniformly distributed random variable on $[0,1]$.

I am interested in the following question: is it possible to find a sequence of random variables, $X_n$, with smooth pdf supported $[-1,1]$ such that the pdfs of $X_n$ converges to the pdf of $Y$?

Edit: The motivation is that this function has shown up in a different non-probabilistic problem. In that context, I am looking for a preferably smooth function approximating $\varphi$. The motivation to find a sequence is that I can use concentration inequalities to sharply bound error estimates.

Answer 1

Let $Z$ have a smooth pdf concenetarted on $(-1,1)$, independent of $Y$ and consider $U_n=1_{|Y| \le 1-\frac 1n }Y+\frac Z n$. Then $U_n$ is absolutely continuous simply because $Z$ is absolutely continuous and independent of $1_{|Y| \le 1-\frac 1n }Y$. Clearly, $U_n \to Y$ a.s. and $|U_n| \le 1$. Since convolution is a smoothing operation, the distribution of $U_n$ as smooth as the distribution of $Z$. This construction does not depend on the specific pdf $\varphi$.

Answer 2

Instead of the sequence given in @geetha290krm's answer, it may be easier to work with

$$U_n=\frac{n-1}{n}Y+\frac Z n$$

where $Z$ can have any arbitrary smooth distribution on the interval $(-1,1)$, which is independent of $Y$.

In this case, random sampling from $U_n$ is easier. Also, the pdf of $U_n$ is given by

$$f_{U_n(x)}=\int_{-1}^{1}f_{\frac Z n}(x-t) \text{d}F_{\frac{n-1}{n}Y}(t) \quad x \in [-1,1].$$