Dirac delta function, is not exactly a function, but with a little abuse of notation you can consider $\delta_{x_0}(x)$ to be a function getting a non-zero value only at point $x=x_0$ and being a probability distribution, i.e. $\int\limits_{-\infty}{+\infty}=1$. Now you can extend this definition to multi-dimensional case with the same requirements. This is the case for Doucet et. al. tutorial. They are estimating the main distribution by a union of multi-dimensional dirac functions.

Dirac delta function, is not exactly a function, but with a little abuse of notation you can consider $\delta_{x_0}(x)$ to be a function getting a non-zero value only at point $x=x_0$ and being a probability distribution, i.e. $\int_{-\infty}^{+\infty}=1$. Now you can extend this definition to multi-dimensional case with the same requirements. This is the case for Doucet et. al. tutorial. They are estimating the main distribution by a union of multi-dimensional dirac functions.