The likelihood is $L(\theta ;x)={\prod }_{i=1}^{n}f(x_i;\theta )$, and a maximum likelihood estimator is any maximizer $\hat{\theta }\in \arg {\max }_{\theta }L(\theta ;X)$. Write $\ell =\log L$ and score $U={\nabla }_{\theta }\ell$. Fisher information for the sample iswhere the equality uses the usual differentiability and interchange assumptions. For independent identically distributed observations $I_n=n{I}_1$. Under standard interior, identifiability and regularity assumptions,Without those assumptions the large-sample normal assertion need not hold, for example at a boundary parameter.

If $\lambda P=\lambda$, induction gives $\lambda P^n=\lambda$ for all $n\ge 0$. By (c), the marginal distribution at time $n$ is consequently $\boxed{\lambda }$. Such a probability vector is a stationary distribution.

Conditioning on the initial state givesThe row-vector convention is important: the initial distribution multiplies the transition matrix on the left.

At $n=0$ the transition probability is ${\delta }_{ij}=(P^0{)}_{ij}$. If the assertion holds at $n$, condition on $X_n=k$ and use the Markov property:Induction proves $\boxed{P(X_n=j\mid X_0=i)=(P^n{)}_{ij}}$. These countable sums contain nonnegative terms and are justified by total probability.

Use $U=Y$, $V=X{Y}^2$, with inverse $y=u$, $x=v/u^2$. The Jacobian is $u^{-2}$, and the transformed support is $0<u<1$, $0<v<u^2$. Thus, for $0<v<1$,Therefore $\boxed{f_V(v)=(v^{-1/2}-1)1_{(0,1)}(v)}$. The integral over its support is $2-1=1$, checking normalization; there are no endpoint atoms.

A $C^1$ embedded hypersurface $M\subseteq {\mathbb{R}}^n$ is locally carried by a $C^1$ coordinate diffeomorphism to an open subset of ${\mathbb{R}}^{n-1}\times \{0\}$. Equivalently, near each of its points it has the form $F^{-1}(0)$ for a $C^1$ function with nonzero gradient there.

The Lagrange multiplier theorem states that if a differentiable function $g$ has a local extremum on such a hypersurface at $x_0$, then $dg(x_0)$ annihilates the tangent space. In a regular local level-set description this isfor some real $\lambda$. The regularity condition $\nabla F(x_0)\ne 0$ is essential; it is what identifies the normal line and permits the local constraint chart.

Differentiability at $x_0$ means that a linear map $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ exists such that $f(x_0+h)=f(x_0)+Lh+o(\Vert h\Vert )$. Since finite-dimensional linear maps are bounded, the right side tends to $f(x_0)$ as $h\to 0$, proving continuity.

For the specified scalar function, $0\le f(x,y)\le x^2\le x^2+y^2$. Dividing by $\sqrt{x^2+y^2}$ gives a bound tending to zero at the origin. Consequently $\boxed{df(0,0)=0}$ and the function is differentiable there.

For the final product assertion, write $f(x_0+h)=Lh+r(h)$, where $r(h)=o(\Vert h\Vert )$, and $g(x_0+h)=g(x_0)+s(h)$ with $s(h)=o(1)$. Since $f(x_0)=0$,Cauchy–Schwarz and boundedness of $g$ near $x_0$ justify the last estimate. Thus $\boxed{dF(x_0)h=⟨df(x_0)h,g(x_0)⟩}$; differentiability of $g$ is unnecessary because $f(x_0)=0$.