= Fréchet limit for a reciprocal integer tail
{c}
{title2=$\mathbb P(M_n/n\leq x)\longrightarrow e^{-1/x},\quad x>0$}
Let $X_i$ be <independent and identically distributed random variables> taking positive integer values with $\mathbb P(X_i\leq k)=k/(k+1)$, and put $M_n=\max_{i\leq n}X_i$. For $x>0$, <independence> gives
$$
\mathbb P(M_n/n\leq x)=\left(1-\frac1{\lfloor nx\rfloor+1}\right)^n\longrightarrow e^{-1/x}.
$$
Indeed $n/(\lfloor nx\rfloor+1)\to1/x$ and $\log(1-u)/u\to-1$ as $u\downarrow0$. For $x\leq0$ the <probability> is zero. The limiting <cumulative distribution function> is continuous, including at zero, and is a <Fréchet distribution> with shape one. Its scale grows linearly with sample size even though the individual <random variables> have infinite <expectation>.
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