Connection between subgaussian/subexponential and exponential family I am wondering if there is any relationship between subgaussian/subexponential with (one parameter) exponential family. 
In particular, is there any sub-family density that belongs to both subgaussian/subexponential as well as exponential family? Furthermore, is there any condition to ensure an exponential family also subgaussian/subexponential?
 A: Following Wikipedia, a subgaussian distribution has tails decaying at least as fast as a gaussian. This is ortogonal to being in an exponential family (ef), some ef distributions are subgaussian, others not (like exponential distribution.) The criteria given in the Wikipedia article should be easy to check for ef distributions.
Subexponential see Wikipedia.  There is a relationship: No ef distributions are subexponential! Subexponential random variables are a special case of heavy-tailed rv's, which also is excluded from exponential families, since they do not have finite moment generating functions for argument $t>0$.  
A: I think there is no direct connection between the two concepts.
sub-gaussian and sub-exponential are definitions which describe the tails of random variables (see discussion below).
The exponential family is a way to parametrize a class of probability distributions:
$$
p(x|\theta)=h(x)\exp(\eta(\theta)T(x)+A(\theta))
$$
The function $h$ will starkly influence the tail of the random variable.
To give a trivial example, let $\eta\equiv 0$ and $A\equiv 0$.
Then, you are left with $p(x|\theta)=h(x)$, with no restrictions but $h\geq 0$.
So, you can make the tail look however you want, by choosing an according $h$.
This does not rule out that some connections can be found.
For example, if $h$ has compact support, so does $p$.
Hence, all elements of this exponential family are sub-gaussian, and consequentially sub-exponential (according to definition 1 below).
On the other hand, knowing that a family of distributions is sub-gaussian or sub-exponential does not tell you at all how they are parametrized.
It is easy to see that these two classes of random variables do not form an exponential family, as they contain random variables with different support.

The answer also depends on your definition of "sub-exponential".
Definition 1 Vershynin defines a random variable $X$ as sub-exponential, if for all $t\geq0$ there is a constant $K>0$ such that:
$$
\mathbb{P}[|X|\geq t]\leq 2\exp(-t/K)
$$
It follows from Propositions 2.5.2. and 2.7.1. in his book that all sub-gaussian random variables are sub-exponential.
Definition 2 The Wikipedia article defines a random variable $X$ supported on the positive part of the real line as sub-exponential, if its cdf $F$ satisfies:
$$
\overline{F^{*2}}(x)\sim 2\overline{F}(x)
$$
as $x\rightarrow\infty$. Here $\overline{F}(x)=1-F(x)$ and $\overline{F^{*2}}$ is the cdf of the sum of two independent copies of $X$.
This definition is not carefully written (at the time at which I'm writing this comment), as it does not rule out discrete random variables.
So, the conclusion in the article "All subexponential distributions are long-tailed," is erroneous, at least with the provided definition.
A better source is this article, where it is assumed that $F(x)<1$ for all $x$, ruling out the discrete counterexample.
