number of ones with neighbours in a random binary string

Question

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts the number of ones having a one on their right. For example, in $N(00101110)=2$ and $N(00101010)=0$.

How to characterize the probability distribution of $N$ as $L\rightarrow \infty$? At least I would like to understand the behaviour of its expectation...

Answer 1

Partial Answer.

Note that $N=\sum_{k=1}^{L-1}\xi_k\xi_{k+1}$, hence $$\mathbb{E}(N)=(L-1)\mathbb{E}({\xi_1\xi_2})=(L-1)p^2.$$