Consistency of kernel density estimator with constant bandwidth

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = \frac{1}{n}\sum_{i=1}^n K_h (x - x_i) \quad = \frac{1}{nh} \sum_{i=1}^n K\Big(\frac{x-x_i}{h}\Big), \end{align} where $K$ a symmetric non-negative function that integrates to one.

All consistency proofs (1,2) require that $h(n) \to 0$ for $n \to \infty$ and I think remember that this is a necessary condition for strong and even weak consistency. Are there any analysis about what happens when the bandwidth $h$ is a constant? Maybe error some bounds on the convergence to $f$?

• Isn't this a duplicate of the question you asked earlier: math.stackexchange.com/questions/914789/… – user76844 Sep 2 '14 at 13:21
• They are similar. The question earlier was about consistency in general and the got correctly answered. Here I am explicitly interested in a constant bandwidth. – Manuel Schmidt Sep 2 '14 at 19:32

Let $X_i \sim \mathcal{N}(\mu,1),\;\;\; \hat \mu = \frac{1}{n}\sum_{i=1}^N X_i + 50$