Is a mixture distribution of gaussian subgaussian?
We know that if $X$ follows a normal distribution $\mathcal{N} (0, \sigma^2)$, then it is automatically $\sigma$-subgaussian by definition.
Suppose we have a mixture of normal random variables $X_1, X_2$. Then does the mixture of these two also have subgaussianity? Suppose the mixture has distribution $f_x(x)$, then:$$f_X(x) = \alpha f_1(x) + (1-\alpha)f_2(x),$$
where $\alpha$ is the probability assigned to a component.
Subgaussianity: Suppose random variable $X$ follows the inequality below: $$\mathbf{\mathbb{E}[e^{\lambda X}]\leq \exp(\lambda^2 \sigma^2)},$$then we say that $X$ is $\boldsymbol{\sigma}$-subgaussian.
Best Answer
Let $X_1$ and $X_2$ be zero mean and sub-Gaussian.
Let $Z$ be such that $P(Z=1)=\alpha$ and $P(Z=2) = 1-\alpha$, and let $X=\mathbf{1}_{Z=1} X_1 + \mathbf{1}_{Z=2}X_2$. Note that $E[X]=0$ as well.
$$E[e^{\lambda X}] = E[E[e^{\lambda X} \mid Z]] = \alpha E[e^{\lambda X_1}] + (1-\alpha) E[e^{\lambda X_2}] \le \alpha e^{\lambda^2 \sigma_1^2/2} + (1-\alpha) e^{\lambda^2 \sigma_2^2/2} \le e^{\lambda^2 \max\{\sigma_1^2, \sigma_2^2\}/2}.$$