Let $J_t$ be a standard Brownian motion, let $X = \{t : J_t = 0\}$ denote the zero set, and let $I(j, n)$ denote the indicator function of the event$$\left\{\text{there exists }s \in \left[{{j-1}\over{n}}, {j\over{n}}\right] \text{ with }J_s = 0\right\}.$$Let$$K_n = \sum_{j=1}^n I(j, n).$$Observe that $K_n$ denotes the number of intervals of the form $\left[{{j-1}\over{n}}, {j\over{n}}\right]$ needed to cover $X \cap [0, 1]$.

- What is the constant $C$ such that$$\lim_{n \to \infty} n^{-1/2} \textbf{E}(K_n) = C?$$
- Is there a constant $C < \infty$ such that for all $n$,$$\textbf{E}[K_n^2] \le C(\textbf{E}[K_n])^2?$$

My apologies, I need these two results for my research, and I am not an expert at probability...