Rescaling function for probability of $k$ adjacent ones in a binary string

Question

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. Denote by $P(k)$ the probability that the string contains at least $k$ adjacent ones. Consider a function $f(n) : \mathbb{N} \rightarrow \mathbb{R}$. What should be the dependence of $f(n)$ on $n$ in order the probability $P(\frac{k}{f(n)})$ to converge to a non-degenerate density as $n \rightarrow \infty$?

Answer 1

Let me try to reformulate the question:

Consider a sequence $(X_k)$ i.i.d. with $P(X_1=1)=p$. Give some asymptotics of $L_n$ the length of the maximal run of ones before time $n$ when $n\to\infty$.

Thus $L_n\geqslant\ell$ if there exists $\ell$ consecutive ones in the sequence $(X_k)_{1\leqslant k\leqslant n}$. The asymptotics of $(L_n)$ are that $$ L_n/\log n\to-\log p, $$ in probability. Furthermore, the normalized version of $L_n$, defined as $$ M_n=L_n+\log p\cdot\log n, $$ converges in distribution to a random variable $M$ with a double exponential distribution, that is, for every real number $x$, $$ \lim\limits_{n\to\infty}P(M_n\leqslant x)=P(M\leqslant x)=\mathrm e^{-(1-p)p^x}. $$ In other words, $M$ has the density $f_M$ defined by $$ f_M(x)=-(1-p)\log p\cdot\mathrm e^{x\log(1-p)-(1-p)\mathrm e^{x\log p}}. $$ Is this what you are after?