Let be $X_1, X_2,\; \ldots $ independent and identically distributed (i.i.d.) random variables with common distribution: $\mathbb{P}(X_k=1)=p,\; \mathbb{P}(X_k=0)=1-p $. We fix a $\lambda > 1$ parameter and let $A_k^{(\lambda)}$ denote the following events: $k=1, 2, 3, \dots $ $ A_k^{(\lambda)}: = \{ \exists r \in [[\lambda^k], [\lambda^{k+1}]-1]\cap\mathbb{N}: X_r=X_{r+1}= \dots =X_{r+k-1}=1\} $ where $[\lambda^k]$ denotes the floor. I.e.: $A_k^{(\lambda)} $ event means that between $[\lambda^k]$a and $[\lambda^{k+1}]-1$ exists a $k$ long run of ones. Let's prove:
a) If \lambda < p^{-1} then $A_k^{(\lambda)} $ events occurs only for finitely many k with probability of 1.
b) If $\lambda > p^{-1}$ then $A_k^{(\lambda)} $ events occurs infinitely many times with probability of 1.
c) What happens when $\lambda=p^{-1}$?
Any help would be appreciated.