Degenerate Distribution in Probability Theory
Definition
For the Location $a$, the following probability density function corresponding to the distribution $\delta_{a}$ is called the degenerate distribution1.
$$ f (x) = \lim_{\sigma \to 0} \exp \left( - \frac{(x - a)^{2}}{2 \sigma^{2}} \right) = \delta (x - a) $$ Here, $\delta$ without a subscript denotes the Dirac delta function.
Explanation
In simple terms, a degenerate distribution is the distribution of a random variable that, in the strict sense, cannot be called a probability distribution: all probability mass is concentrated at a single point, randomness disappears, the event becomes certain, and it is treated as a kind of constant function.
For example, in gambling, a loaded die that always shows $6$ has the degenerate distribution $\delta_{6}$. Of course there can be the experiment of throwing the die, and one can formally describe a probability distribution, but no matter how it is thrown, the outcome is already determined to be $6$.
Degenerate distributions are often mentioned as a basic assumption in the definition of stable distributions.