Erd\H{o}s-R\'enyi Law

 

Let $0<p=1-q<1$ and $X_1,X_2,\ldots$ be an i.i.d. Bernoulli sequence with $p=\mathbb{P}(X_i=1)=1-\mathbb{P}(X_i=0)$. Denote by $S_n$ the length of the longest consectutive run of heads (i.e., $1$'s) within the first $n$ tosses. Erd\H{o}s-R\'enyi Law tells us the asymptotic behaviors of $S_n$: almost surely,

$$\lim\limits_{n\to\infty}\frac{S_n}{\log_{1/p}n}=1.$$

 

See  1. Erd\H{o}s-R\'enyi, On a new law of large number, J. Anal. Math., 1970. 2. Mao, Wang and Wu, Large deviation behavior for the longest head run in an IID Bernoulli sequence. J Theor Probab, 2015.