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Let \(\mu \) be a measure on a real inner product space \(E\). The characteristic function of \(\mu \), denoted by \(\hat{\mu }\), is the function \(E \to \mathbb {C}\) defined by

The characteristic function of a random variable \(X\) is defined as the characteristic function of \(\mathcal L(X)\).

We say that \(X_n\) converges in distribution to \(X\) and write \(X_n \xrightarrow {d} X\) if \(\mathcal L(X_n) \xrightarrow {w} \mathcal L(X)\).

A measure on \(\mathbb {R}\) is Gaussian if it is equal to \(\mathcal{N}(m, \sigma ^2)\) for some \(m \in \mathbb {R}\) and \(\sigma ^2 \ge 0\), in which \(\mathcal{N}(m, \sigma ^2)\) is the measure absolutely continuous with respect to the Lebesgue measure with density \(x \mapsto \frac{1}{\sqrt{2 \pi \sigma ^2}}e^{- \frac{1}{2\sigma ^2}(x - m)^2}\) if \(\sigma ^2{\gt}0\) and the Dirac probability measure at \(m\) if \(\sigma ^2 = 0\).

A Borel measure \(\mu \) on a separable Banach space \(E\) is said to be Gaussian if for all continuous linear maps \(\ell : E \to \mathbb {R}\), the pushforward \(\ell _*\mu \) is a real Gaussian measure.

A set \(\mathcal F\) of functions \(E \to F\) separates points in \(E\) if for all \(x, y \in E\) with \(x \ne y\), there exists \(f \in \mathcal F\) with \(f(x) \ne f(y)\).

A set \(S\) of measures on \(\Omega \) is tight if for all \(\varepsilon {\gt} 0\) there exists a compact set \(K\) such that for all \(\mu \in S\), \(\mu (K^c) \le \varepsilon \).

In a topological space \(S\) in which the opens are measurable, we say that a sequence of probability measures \(\mu _n\) converges weakly to a probability measure \(\mu \) and write \(\mu _n \xrightarrow {w} \mu \) if \(\mu _n[f] \to \mu [f]\) for every bounded continuous function \(f : S \to \mathbb {R}\).

The sum of two independent real Gaussian random variables is Gaussian.

In a Borel space \(E\), if two probability measures \(\mu , \nu \) on \(E\) are such that \(\mu [f] = \nu [f]\) for all \(f \in C_b(E, \mathbb {C})\), the set of bounded continuous functions from \(E\) to \(\mathbb {C}\), then \(\mu = \nu \).

If two random variables \(X, Y : \Omega \to S\) with laws \(\mu \) and \(\nu \) are independent, then \(X+Y\) has characteristic function \(\hat{\mu } \times \hat{\nu }\).

For \(\mu \) a probability measure on an inner product space \(E\) and \(r {\gt} 0\), \(a \in E\),

For \(\mu \) a probability measure on \(\mathbb {R}\) and \(r {\gt} 0\),

For all \(t\), \(\Vert \hat{\mu }(t)\Vert \le 1\).

If \(\mu [|X|^n] {\lt} \infty \) then the characteristic function of \(\mu \) is continuously differentiable of order \(n\).

The characteristic function is a continuous function.

\(\hat{\mu }(-t) = \overline{\hat{\mu }(t)}\).

For \(a \in \mathbb {R}\) and \(X\) a random variable with law \(\mu \), the characteristic function of \(a X\) is \(t \mapsto \hat{\mu }(at)\).

Let \(X\) be a real random variable with characteristic function \(\phi \), with \(\mathbb {E}[\vert X \vert ^n] {\lt} \infty \). Then as \(t \to 0\),

Let \(\mathcal A\) be a subalgebra of \(C_b(E, \mathbb {C})\) that separates points. Let \(\mu , \mu _1, \mu _2, \ldots \) be probability measures, with \((\mu _n)\) tight. If for all \(f \in \mathcal A\), \(\mu _n[f] \to \mu [f]\), then \(\mu _n \xrightarrow {w} \mu \).

Let \(X\) be a real random variable with characteristic function \(\phi \) with \(\mathbb {E}[\vert X \vert ^n] {\lt} \infty \) and let \(k \le n\). Then the \(k\)^th derivative of \(\phi \) is \(\phi ^{(k)}(t) = \mathbb {E}[i X]^k e^{i t X}\)   .

Let \(\mu , \nu \) be two finite measures in a standard Borel space \(E\). Let \(\mathcal{A}\) be a subalgebra of \(C_b(E, \mathbb {R})\) that separates points. If for all \(g \in \mathcal A\), \(\mu [g] = \nu [g]\), then for all \(f \in C_b(E, \mathbb {R})\) and all \(\varepsilon {\gt} 0\),

\(x \mapsto (t \mapsto \exp (i \langle t, x \rangle ))\) is a monoid homomorphism from \((E,+)\) to \(C_b(E, \mathbb {C})\).

Two finite measures on a complete separable metric space are equal iff they have the same characteristic function.

The Gaussian distribution \(\mathcal N(m, \sigma ^2)\) has characteristic function \(\phi (t) = e^{itm - \sigma ^2 t^2 /2}\).

A finite collection of random variables are independent iff their joint law is the product of their respective laws.

Let \(\mu \) be a finite measure on a Borel space \(E\) and let \(K\) be a measurable compact set such that \(\mu (K^c) \le \varepsilon \). Let \(f \in C(E, \mathbb {R})\). Then

For any \(f \in C_b(E, \mathbb {R})\) and a probability measure \(\mu \),

The sum of two independent Gaussian random variables is Gaussian.

The real Gaussian measures are Gaussian measures in the sense of Definition 5.5.

A real Gaussian measure is a probability measure.

A Gaussian measure is a probability measure.

Let \((\mu _n)_{n \in \mathbb {N}}\) be a sequence of measures on a finite-dimensional inner product space \(E\). Then the set \(\{ \mu _n \mid n \in \mathbb {N}\} \) is tight iff for all \(y \in E\), \(\lim _{r \to +\infty }\limsup _{n \to \infty } \mu _n\{ \vert \langle y, x\rangle \vert {\gt} r\} = 0\).

Let \(S\) be a set of measures on a finite-dimensional normed space \(E\). Then \(S\) is tight iff \(\lim _{r \to +\infty }\sup _{\mu \in S} \mu \{ \Vert x \Vert {\gt} r\} = 0\).

Let \(S\) be a set of measures on a finite-dimensional inner product space \(E\). Then \(S\) is tight iff for all \(y \in E\), \(\lim _{r \to +\infty }\sup _{\mu \in S} \mu \{ \vert \langle y, x\rangle \vert {\gt} r\} = 0\).

Let \(E\) be a metric space and let \(S \subseteq \mathcal P(E)\) be tight. Then the closure of \(S\) is compact.

The star-subalgebra \(\mathcal M\) separates points.

The functions of the form \(x \mapsto \sum _{k=1}^n a_k e^{i\langle t_k, x\rangle }\) for \(n \in \mathbb {N}\), \(a_1, \ldots , a_n \in \mathbb {R}\) and \(t_1, \ldots , t_n \in E\) are a star-subalgebra of \(C_b(E, \mathbb {C})\).

Let \(E\) be a finite dimensional real inner product space and let \(b_1, \ldots , b_d\) be an orthonormal basis of \(E\). Let \(X_1, \ldots , X_d\) be independent standard Gaussian random variables on \(\mathbb {R}\). Then the law of \(X_1 b_1 + \ldots + X_d b_d\) is a Gaussian measure on \(E\).

For a \(n\)-times continuously differentiable function \(f : \mathbb {R} \to E\), for any \(x_0\in \mathbb {R}\),

Let \(\mu , \mu _1, \mu _2, \ldots \) be probability measures on \(\mathbb {R}^d\) with characteristic functions \(\hat{\mu }, \hat{\mu }_1, \hat{\mu }_2, \ldots \) such that \(\hat{\mu }_n \to \hat{\mu }\). Then for all exponential polynomial \(f \in \mathcal M\), \(\mu _n[f] \to \nu [f]\).

For \(t\in \mathbb {C}\), \(\lim _{n\to \infty }(1+t/n+o(1/n))^n=\exp (t)\) (where the little-\(o\) term may be complex).

Let \((\mu _n)_{n \in \mathbb {N}}\) be measures on \(\mathbb {R}^d\) with characteristic functions \((\hat{\mu }_n)\). If \(\hat{\mu }_n\) converges pointwise to a function \(f\) which is continuous at 0, then \((\mu _n)\) is tight.

Let \(\mu , \mu _1, \mu _2, \ldots \) be probability measures on \(\mathbb {R}^d\) with characteristic functions \(\hat{\mu }, \hat{\mu }_1, \hat{\mu }_2, \ldots \). Then \(\mu _n \xrightarrow {w} \mu \) iff for all \(t\), \(\hat{\mu }_n(t) \to \hat{\mu }(t)\).

Let \(X_1, X_2, \ldots \) be i.i.d. real random variables with mean 0 and variance 1, and let \(Z\) be a random variable with law \(\mathcal N(0,1)\). Then

Let \(X, X_1, X_2, \ldots \) be random variables in \(\mathbb {R}^d\). Then \(X_n \xrightarrow {d} X\) iff for every \(a \in \mathbb {R}^d\), \(\langle a, X_n \rangle \xrightarrow {d} \langle a, X \rangle \).

Let \(X_1, X_2, \ldots \) be i.i.d. random variables in \(\mathbb {R}^d\) with mean 0 and identity covariance matrix, and let \(Z\) be a random variable with law \(\mathcal N(0,I_d)\). Then

Let \(E\) be a complete separable metric space and let \(S \subseteq \mathcal P(E)\). Then the following are equivalent:

1. \(S\) is tight.

2. the closure of \(S\) is compact.

Let \(E\) be a complete separable pseudo-metric space. Let \(\mathcal A \subseteq C_b(E, \mathbb {C})\) be a star-subalgebra that separates points. Then if two probability measures \(\mu , \nu \) of \(E\) are such that \(\mu [f] = \nu [f]\) for all \(f \in \mathcal A\), then \(\mu = \nu \).