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I am looking for a nontrivial example of a singular distribution that when convolved with a Gaussian distribution has a pdf of a 'simple' form.

I let 'simple' be something that you interpret yourself.

Singular distributions are an import class of distributions that are often 'swept under the carpet.' I would like to see a nice illustrative example of how to work with such distributions.

One way to do this is to give a characteristic function $\phi(t)$ that when multiplied by $e^{-t^2/2}$ has a simple Fourier inverse.

However, I don't have a good choice of the characteristic function $\phi(t)$ that would lead to a meaningful result.

For example, for the Cantor distribution, the characteristic function is given by \begin{align} \phi(t)=e^\frac{it}{2} \prod_{i=1}^\infty \cos \left( \frac{t}{3^k} \right). \end{align} However, it and is not easy to work with this characteristic function \begin{align} \phi(t) e^{-\frac{t^2}{2}}. \end{align} In particular, it is difficult to fuind its Fourier inverse.

Edit: By singular distributions I mean: A singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.

Edit 2: Another approach we can take is to look at the convolution directly. That is look at the $U=X+V$ where $V$ is has a singular distribution and $X$ is Gaussian, in this case, the pdf of $U$ is given by \begin{align} f_U(u)=E\left[ \frac{1}{\sqrt{2 \pi}} e^{-\frac{(u-V)^2}{2}} \right]. \end{align}

I was wondering if we can come up with a sequence of random variables $V_n$ that converges in distribution to some $V$ with a singular distribution, for which we can compute the limit \begin{align} \lim_{ n \to \infty}E\left[ \frac{1}{\sqrt{2 \pi}} e^{-\frac{(u-V_n)^2}{2}} \right]. \end{align}

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  • $\begingroup$ Am I right in saying that the Dirac delta distribution is a singular distribution? If so, it's convolution with a Gaussian, when normalised, would be a Dirac delta distribution. $\endgroup$ – Damien May 17 '18 at 14:45
  • $\begingroup$ Hi, @DanielBeale I added a definition of what I mean by singular distribution. $\endgroup$ – Boby May 17 '18 at 17:45
  • $\begingroup$ Sorry I misunderstood the "where the probability of each point in that set is zero" part of your definition. I've only seen this referred to as a "singularly continuous measure" in the past, but I see Wikipedia uses that same definition... $\endgroup$ – fourierwho May 21 '18 at 22:39
  • $\begingroup$ @fourierwho No problem. Most likely, it is difficult to find a good example. Here is an interesting paper that I found this topic it might be of interest to you jstor.org/stable/pdf/… $\endgroup$ – Boby May 22 '18 at 0:18
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    $\begingroup$ @Boby As I understand it, the distribution $\ \mu\ $ you're looking for should have the property that $\ \mu(A)=1\ $ for some set $\ A\ $ of Lebesgue measure zero, and $\ \mu(\{a\})=0\ $ for every $\ a\in A\ $. Is that correct? $\endgroup$ – lonza leggiera Aug 22 at 6:38
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I may be taking a very liberal interpretation of "simple", but hear me out. Define \begin{align*} Y_n = 2\sum_{k=1}^{n}\frac{X_k}{3^k} \end{align*} where $X_1, X_2, \cdots \overset{\text{iid}}{\sim} \text{Bernoulli}(\frac{1}{2})$. It's a well-known result that $Y_n$ converges in distribution to a Cantor random variable. Trivially, \begin{align*} \mathbb{P}(Y_n = c) = \begin{cases} \frac{1}{2^n} & c \in S_n \overset{\text{def}}{=}\{2\sum_{k=1}^{n}\frac{x_k}{3^k}: x_1, \cdots, x_n \in \{0, 1\}\} \\ 0 & \text{otherwise} \end{cases} \end{align*} For a discrete random variable $D$ taking values $(d_1, \cdots, d_n)$ with probabilities $(p_1, \cdots, p_n)$, the convolution of $D+Z$ for a standard Gaussian $Z$ is a mixture of Gaussians with density \begin{align*} f_{D+Z}(t) = \sum_{k=1}^{n}p_k\phi(t - d_k) \end{align*} where $\phi(t) = (2\pi)^{-1/2}e^{-t^2/2}$ is the density of a standard Gaussian. So \begin{align*} f_{Y_n + Z}(t) = \frac{1}{2^n}\sum_{c \in S_n}\phi(t-c) \end{align*} Checking all necessary conditions and taking the limit $n \rightarrow \infty$, we have the convolution of a Cantor and a standard Gaussian. So, this convolution can be interpreted as an (uncountably) infinite mixture of Gaussians. This isn't quite satisfactory because an infinite mixture of Gaussians can pretty much approximate any continuous distribution, and therefore the answer is still "something", but at least this way we can have a constructive approximation.

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