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For ${\chi }_0={\varphi }_0$ and ${\chi }_n=\sqrt{{\varphi }_{n-1}{\varphi }_n}$, the sequence $({\chi }_n)$ is asymptotic, but ${\chi }_n/{\varphi }_n\to \infty$ for every $n\ge 1$.

For a positive asymptotic sequence $({\varphi }_n)$, define ${\psi }_0={\varphi }_0$ andThen ${\psi }_n\sim {\varphi }_n$ term by term and $({\psi }_n)$ is again an asymptotic sequence.

For an independent sample from $\mathrm{Uniform}[-\theta ,\theta ]$, the maximum likelihood estimator is $M_n={\max }_i|X_i|$ andThe parameter-dependent support makes the convergence rate $n$ rather than $\sqrt{n}$, so the regular maximum-likelihood central limit theorem does not apply.

If $\sqrt{n}(T_n-\theta )$ is asymptotically normal, differentiability gives $\sqrt{n}(g(T_n)-g(\theta ))$ the variance multiplied by $g^{\prime }(\theta {)}^2$.

For estimating equations, asymptotic covariance often has the form $A^{-1}B{A}^{-1}$, called a sandwich covariance.

Regression heteroskedasticity-consistent standard errors use a specific sandwich covariance estimate; general estimating equations can produce sandwich covariances outside regression.

Slutsky's theorem combines convergence in distribution with convergence in probability through continuous algebraic operations.

Forrepeated integration by parts gives

For fixed $x$ and $y\to \infty$,

If an algebra is defined over a parameter ring $R$, a ring homomorphism $R\to k$ produces a specialized $k$-algebra by extension of scalars. Defining relations specialize by replacing every parameter by its image in $k$.

A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

A left ideal $L$ of an associative algebra $A$ is a vector subspace satisfying $AL\subseteq L$. Equivalently, it is a submodule of the left regular $A$-module.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal. Equivalently, it is Artinian exactly when its Krull dimension is zero.

The Condorcet paradox is a cycle in collective pairwise preferences even though each individual voter's preference is transitive.

A set is in ${\Pi }_2$ when membership can be written with one universal block followed by one existential block over a computable predicate. Equivalently, in the word-function formulation, there is a partial computable $f$ such that

A set $A$ is in ${\Sigma }_1$ exactly when it is the domain of a partial computable function. Equivalently, there is a computable predicate $R$ such that

A ${\Sigma }_1$ formula is equivalent to $\exists \mathbf{y}\delta$ with $\delta$ bounded. A ${\Pi }_1$ formula is equivalent to $\forall \mathbf{y}\delta$ with $\delta$ bounded.

If $g$ is holomorphic and nonvanishing on a punctured disc, thenis the integer winding number of $g(C)$ about zero. If it equals $k$, the logarithmic derivative of $z^{-k}g(z)$ has a removable singularity at zero.

If $f,g$ are holomorphic near the closure of a bounded domain and $|g|<|f|$ on its boundary, then $f$ and $f+g$ have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy $f+tg$.

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.