= Solution
A <non-degenerate probability distribution> is not concentrated at a single point. In <extreme value theory>, a non-degenerate <distribution function> $F$ is <max-stable> if, for each integer $n\geq1$, constants $a_n>0,b_n\in\mathbb R$ satisfy $F(a_nx+b_n)^n=F(x)$ for every $x$. This says that a normalized <sample maximum> has the same law as one observation. A <distribution function> $F$ belongs to the <maximum domain of attraction> of a non-degenerate $G$ if there are $a_n>0,b_n$ such that
$$
F(a_nx+b_n)^n\longrightarrow G(x)
$$
at every <continuity> point of $G$. Equivalently, $(M_n-b_n)/a_n$ has <convergence in distribution> to $G$.
The <extremal types theorem> says that any such non-degenerate limit is <max-stable> and, up to a positive affine change of variable, is exactly one of the following:
$$
\begin{aligned}
\Lambda(x)&=e^{-e^{-x}},&&x\in\mathbb R,\\
\Phi_\alpha(x)&=e^{-x^{-\alpha}},&&x>0,\quad \Phi_\alpha(x)=0\ (x\leq0),\\
\Psi_\alpha(x)&=e^{-(-x)^\alpha},&&x<0,\quad \Psi_\alpha(x)=1\ (x\geq0),
\end{aligned}
\qquad\alpha>0.
$$
These are respectively the <Gumbel distribution>, <Fréchet distribution>, and <negative Weibull distribution>. The theorem is also known as the <Fisher–Tippett–Gnedenko theorem>.
Useful sufficient conditions can be expressed using the <survival function> $\overline F=1-F$ and the <right endpoint of a distribution> $x_F$. An infinite endpoint with $\overline F(tx)/\overline F(t)\to x^{-\alpha}$ for every $x>0$ gives the <Fréchet distribution> domain. A finite endpoint with $\overline F(x_F-sx)/\overline F(x_F-s)\to x^\alpha$ as $s\downarrow0$ gives the <negative Weibull distribution> domain. For the <Gumbel distribution>, it suffices that a positive <Gumbel auxiliary function> $a(t)$ gives $\overline F(t+a(t)x)/\overline F(t)\to e^{-x}$ for every real $x$ as $t\uparrow x_F$. These are <regular variation> and exponential tail-ratio conditions; they need no proof here.
In case (i), $\overline F_1(x)=(1-x)^2$ on $0<x<1$, so the finite-endpoint ratio is exactly $x^2$. Therefore \b[the domain is negative Weibull with shape $2$]. With $a_n=n^{-1/2}$ and $b_n=1$,
$$
\mathbb P\{\sqrt n(M_n-1)\leq x\}
=\left(1-\frac{x^2}{n}\right)^n\longrightarrow e^{-x^2}\quad(x<0),
$$
and the limit is one for $x\geq0$.
In case (ii), $\overline F_2(t)=e^{-\lambda t}$ has exponential tail ratio with the constant <Gumbel auxiliary function> $1/\lambda$. Therefore \b[the domain is Gumbel]. Choosing $a_n=1/\lambda$ and $b_n=\log n/\lambda$ gives
$$
\mathbb P\{\lambda M_n-\log n\leq x\}
=\left(1-\frac{e^{-x}}n\right)^n\longrightarrow e^{-e^{-x}}.
$$
In case (iii), $\overline F_3(t)=(1+t)^{-\lambda}$ has <regular variation> of index $-\lambda$, so \b[the domain is Fréchet with shape $\lambda$]. The convenient choices $a_n=n^{1/\lambda}$ and $b_n=-1$ give
$$
\mathbb P\{(M_n+1)/n^{1/\lambda}\leq x\}
=\left(1-\frac{x^{-\lambda}}n\right)^n\longrightarrow e^{-x^{-\lambda}}\quad(x>0),
$$
with limit zero for $x\leq0$.
Finally, define the <empirical distribution function> $F_n(x)=n^{-1}\sum_{i=1}^n\mathbf1_{\{X_i\leq x\}}$. On a sample with positive threshold $t=X_{(n-k)}$ and exactly $k$ observations strictly above $t$, its <survival function> has $1-F_n(t)=k/n$. Using the <Tonelli theorem> to integrate the finite nonnegative sum gives
$$
\int_t^\infty\frac{1-F_n(x)}{1-F_n(t)}\frac{dx}{x}
=\frac1k\sum_{i:X_i>t}\int_t^{X_i}\frac{dx}{x}
=\boxed{\frac1k\sum_{j=1}^k\log\frac{X_{(n-j+1)}}{X_{(n-k)}}=\widehat\gamma_H.}
$$
This is the <Hill estimator> as an empirical plug-in version of the tail-integral limit. It does not prove consistency for fixed $k$; that is a separate issue.
There is a finite-sample qualification because the question permits ties and does not assume positive observations. The formula requires $1\leq k<n$ and $t>0$. If $r=\#\{i:X_i>t\}<k$ because the threshold is tied, direct substitution gives
$$
\frac1r\sum_{j=1}^k\log\frac{X_{(n-j+1)}}t\qquad(r>0),
$$
where terms equal to the threshold contribute zero. This is generally $k/r$ times the displayed <Hill estimator>, rather than the same estimator. If $r=0$, the empirical plug-in denominator is zero. For a <continuous probability distribution> the no-tie condition holds almost surely; for a <Fréchet distribution> domain and fixed $k$, the threshold is positive with probability tending to one. Thus \b[the stated plug-in identity holds at a positive untied threshold], and the literal claim for all samples from an arbitrary <distribution function> needs this qualification.
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