Modern Stochastics: Theory and Applications

The sequence ${\mathrm{(}{\mathit{Z}}^{\mathit{n}}\mathrm{)}}_{\mathit{n}\in \mathbb{N}}$ converges weakly in the Skorokhod space $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$ to a standard Brownian motion W as n tends to infinity.

As the distribution of the random vector in (1) is exactly the uniform measure ${\mathit{\mu }}_{\mathit{S}\mathrm{,}\mathit{n}}$, the proof of the convergence of finite-dimensional distributions is in line with the classical Poincaré–Borel lemma: by the law of large numbers, $\frac{1}{\mathit{n}}{\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{X}}_{\mathit{i}}^2\to 1$ in probability. Hence, by the continuous mapping theorem, $\sqrt{\mathit{n}}\mathrm{/}\mathrm{(}\Vert \mathrm{(}{\mathit{X}}_1\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{n}}\mathrm{)}\Vert \mathrm{)}\to 1$ in probability, and, by Donsker’s theorem and Slutsky’s theorem, we conclude the convergence of finite-dimensional distributions.

For the tightness we consider the increments of the process ${\mathit{Z}}^{\mathit{n}}$ and make use of a standard criterion. For all $\mathit{s}\le \mathit{t}$ in $[0\mathrm{,}1]$, we denote Due to the symmetry of the standard n-dimensional normally distributed vector $\mathrm{(}{\mathit{X}}_1\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{n}}\mathrm{)}$, for all pairwise different $\mathit{i}\mathrm{,}\mathit{j}\mathrm{,}\mathit{k}\mathrm{,}\mathit{l}$, we observe Let $\mathit{s}\le \mathit{u}\le \mathit{t}$ in $[0\mathrm{,}1]$. Thus via (3), we conclude and therefore We denote for shorthand ${\mathit{m}}_1:=\lfloor \mathit{n}\mathit{t}\rfloor -\lfloor \mathit{n}\mathit{u}\rfloor$, ${\mathit{m}}_2:=\lfloor \mathit{n}\mathit{u}\rfloor -\lfloor \mathit{n}\mathit{s}\rfloor$ and ${\mathit{m}}_3:=\mathit{n}-\mathrm{(}\lfloor \mathit{n}\mathit{t}\rfloor -\lfloor \mathit{n}\mathit{s}\rfloor \mathrm{)}$. Then we observe for pairwise independent chi-squared random variables ${\mathit{\chi }}_{\mathit{m}}^2$ with m degrees of freedom. We recall that $\frac{{\mathit{\chi }}_{\mathit{m}}^2}{{\mathit{\chi }}_{\mathit{m}}^2+{\mathit{\chi }}_{\mathit{k}}^2}$ is $\text{Beta}\mathrm{(}\mathit{m}\mathrm{/}2\mathrm{,}\mathit{k}\mathrm{/}2\mathrm{)}$-distributed with Hence a computation via (4) yields and therefore

Thus the well-known criterion [1, Theorem 15.6] (cp. Remark 1 in [15]) implies the tightness of ${\mathit{Z}}^{\mathit{n}}$.  □

(i) The heuristic connection of the Wiener measure and the uniform measure on an infinite-dimensional sphere goes back to Norbert Wiener’s study of the differential space, [21]. The first informal presentation of Theorem 1 and further historical notes can be found in [12]. The first rigorous proof is given in Section 2 of [4]. However, the authors make use of nonstandard analysis and the functional ${\mathit{Q}}_{\mathit{n}\mathrm{,}2}$. To the best of our knowledge, the first proof of Theorem 1 is [17]. In contrast, our proof is based on the pretty decoupling in the tightness argument. Moreover, this approach is extended in Section 4 to a simpler proof of Theorem 1 in [3].

(ii) According to the historical comments in [20, Section 2.2], the Poincaré–Borel lemma could be also attributed to Maxwell and Mehler.

Let ${\mathit{X}}_{\mathit{i}}={\mathit{B}}_{\mathit{i}}^{\mathit{H}}-{\mathit{B}}_{\mathit{i}-1}^{\mathit{H}}$, $\mathit{i}\in \mathbb{N}$, be the increments of a fractional Brownian motion ${\mathit{B}}^{\mathit{H}}$. Then, in the Skorokhod space $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$, as n tends to infinity:

Taqqu’s limit theorem implies, for all $\mathit{H}\in \mathrm{(}0\mathrm{,}1\mathrm{)}$, in the Skorokhod space $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$. Then, thanks to (5), we conclude as in Theorem 5.  □

Due to the different correlations between the random variables ${\mathit{X}}_{\mathit{i}}$ in Theorem 6, there is no symmetric and trivial sequence of measures ${\hat{\mathit{\mu }}}_{\mathit{S}\mathrm{,}\mathit{n}\mathrm{,}\mathit{p}}$ on the ${\ell }_{\mathit{p}}^{\mathit{n}}$-spheres and some simple ${\bar{\mathit{Q}}}_{\mathit{n}\mathrm{,}\mathit{p}}$-type pathwise functionals, which represent the distributions of ${\bar{\mathit{Q}}}_{\mathit{n}\mathrm{,}\mathit{p}}\mathrm{(}{\mathit{X}}_1\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{n}}\mathrm{)}$. However, it would be interesting, whether some uniform or surface measures on geometric objects in combination with simple ${\bar{\mathit{Q}}}_{\mathit{p}}$-type pathwise functionals allow similar Donsker-type theorems for fractional Brownian motion or other Gaussian processes?

Assume the notations above and denote Then the following assertions, with n tending to infinity, are equivalent:

The equivalence of $\mathrm{(}\mathit{a}\mathrm{)}$ and $\mathrm{(}\mathit{b}\mathrm{)}$ is the celebrated result [8, Theorem 3]. Since the implications $\mathrm{(}\mathit{d}\mathrm{)}\Rightarrow \mathrm{(}\mathit{c}\mathrm{)}\Rightarrow \mathrm{(}\mathit{b}\mathrm{)}$ are trivial, the proof in [3] is completed by showing $\mathrm{(}\mathit{a}\mathrm{)}\Rightarrow \mathrm{(}\mathit{d}\mathrm{)}$.

Thanks to a tightness argument as in the proof of Theorem 1, we obtain a simpler alternative for the proof.

As stated in the remark, we already know that $\mathrm{(}\mathit{d}\mathrm{)}\Rightarrow \mathrm{(}\mathit{c}\mathrm{)}\Rightarrow \mathrm{(}\mathit{b}\mathrm{)}\iff \mathrm{(}\mathit{a}\mathrm{)}$. We denote By the continuity of the paths of the Brownian motion and [1, Section 18], we obtain the equivalence $\mathrm{(}\mathit{c}\mathrm{)}\iff \mathrm{(}{\mathit{c}}_0\mathrm{)}$. We denote by ${\mathit{d}}_0$ the Skorokhod metric on $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$ which makes it a Polish space. The Skorokhod–Dudley Theorem [9, Theorem 4.30] and $\mathrm{(}{\mathit{c}}_0\mathrm{)}$ imply almost surely on an appropriate probability space. Since the uniform topology is finer than the Skorokhod topology ([1, Section 18]), we conclude assertion $\mathrm{(}\mathit{d}\mathrm{)}$. Thus it remains to prove $\mathrm{(}\mathit{a}\mathrm{)}\Rightarrow \mathrm{(}{\mathit{c}}_0\mathrm{)}$. Firstly we consider finite-dimensional distributions. Due to [8, Lemma 3.2], the sequence ${\mathrm{(}{\mathit{b}}_{\mathit{n}}\mathrm{)}}_{\mathit{n}\in \mathbb{N}}$ with ${\mathit{S}}_{\mathit{n}}\mathrm{/}{\mathit{b}}_{\mathit{n}}\stackrel{\mathit{d}}{\to }\mathcal{N}\mathrm{(}0\mathrm{,}1\mathrm{)}$ fulfills ${\mathit{V}}_{\mathit{n}}\mathrm{/}{\mathit{b}}_{\mathit{n}}\to 1$ in probability and ${\mathit{b}}_{\mathit{n}}=\sqrt{\mathit{n}}\mathit{L}\mathrm{(}\mathit{n}\mathrm{)}$ for some slowly varying at infinity function L. The continuous mapping theorem implies ${\mathit{b}}_{\mathit{n}}\mathrm{/}{\mathit{V}}_{\mathit{n}}\to 1$ in probability. Take arbitrary $\mathit{N}\in \mathbb{N}$, ${\mathit{a}}_1\mathrm{,}\dots \mathrm{,}{\mathit{a}}_{\mathit{N}}\in \mathbb{R}$ and ${\mathit{t}}_1\mathrm{,}\dots \mathrm{,}{\mathit{t}}_{\mathit{N}}\in [0\mathrm{,}1]$. Without loss of generality, we assume ${\mathit{t}}_1\mathrm{<}\cdots \mathrm{<}{\mathit{t}}_{\mathit{N}}$ and denote ${\mathit{t}}_0:=0$ and ${\mathit{t}}_{\mathit{N}+1}:=1$. Then, by the independence of the random variables ${\mathit{S}}_{\lfloor \mathit{n}{\mathit{t}}_{\mathit{i}}\rfloor }-{\mathit{S}}_{\lfloor \mathit{n}{\mathit{t}}_{\mathit{i}-1}\rfloor }$, $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{N}+1$, for every fixed $\mathit{n}\in \mathbb{N}$, Lévy’s continuity theorem and the normality of the random vector $\mathrm{(}{\mathit{Y}}_1\mathrm{,}\dots \mathrm{,}{\mathit{Y}}_{\mathit{N}+1}\mathrm{)}$, we obtain as n tends to infinity. As the sequence ${\mathrm{(}{\mathit{b}}_{\mathit{n}}\mathrm{)}}_{\mathit{n}\in \mathbb{N}}$ is regularly varying with exponent $1\mathrm{/}2$, it is easily seen that Via the continuous mapping theorem, we conclude Slutsky’s theorem implies what means the convergence of finite-dimensional distributions.

The tightness follows again by the criterion [1, Theorem 15.6]. By the identical distribution, for all $\mathit{m}\le \mathit{n}$, we have Thanks to the value 1 on the left hand side in (6) for $\mathit{m}=\mathit{n}$, we conclude In contrast to (3), for possibly nonsymmetric random variables, the Cauchy–Schwarz inequality and [8, (3.10)] yields a constant ${\mathit{c}}_{\mathit{X}}\mathrm{<}\infty$ such that for every $\mathit{r}\in \{2\mathrm{,}3\mathrm{,}4\}$, Applying the estimates in (7) on the terms in (2) gives that Hence, we obtain and the proof concludes as in Theorem 1.  □

(i) By the same reasoning, we obtain Theorem 5 for the sequence of i.i.d. variables $\mathit{X}\mathrm{,}{\mathit{X}}_1\mathrm{,}{\mathit{X}}_2\mathrm{,}\dots$ such that Theorem 8 $\mathrm{(}\mathit{a}\mathrm{)}$ is fulfilled.

(ii) In [2], a similar counterpart of Theorem 8 for α-stable Lévy processes is established. An interesting question would be on a uniqueness result similar to Theorem 5.