The self-normalized Donsker theorem revisited

We extend the Poincar\'{e}--Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Cs\"{o}rg\H{o} et al. (2003). Some notes on spheres with respect to $\ell_p$-norms are given.


Introduction
Let S n−1 (d) = {x ∈ R n : x = d} be the (n − 1)-sphere with radius d, where · denotes the euclidean norm. The uniform measure on the unit sphere S n−1 := S n−1 (1) can be characterized as µ S,n d = (X 1 , . . . , X n ) (X 1 , . . . , X n ) , where (X 1 , . . . , X n ) is a standard n-dimensional normal random variable. The celebrated Poincaré-Borel lemma is the classical result on the approximation of a Gaussian distribution by projections of the uniform measure on S n−1 ( √ n) as n tends to infinity: Let n ≥ m and π n,m : R n → R m be the natural projection. The uniform measure on the sphere S n−1 ( √ n) is given by √ n µ S,n . Then, for every fixed m ∈ N, √ nµ S,n • π −1 n,m converges in distribution to a standard m-dimensional normal distribution as n tends to infinity, cf. [Lifshits (2012), Proposition 6.1]. Following the historical notes in [Diaconis and Freedman (1987), Section 6] on the earliest reference to this result byÉmile Borel, we acquire the usual practice to speak about the Poincaré-Borel lemma. Among other fields, this convergence stimulated the development of the infinite-dimensional functional analysis (cf. [McKean (1973)]) as well as the concentration of measure theory (cf. [Ledoux and Talagrand (1991), Section 1.1]). In particular, it inspired to consider connections of the Wiener measure and the uniform measure on an infinite-dimensional sphere ( [Wiener (1923)]). Such a Donsker-type result is firstly proved in [Cutland and Ng (1993)] by nonstandard methods. For the illustration, we make use of the notations in [Dryden (2005)], where this result is used for statistical analysis of measures on high-dimensional unit spheres. Define the functional converges weakly to a Brownian motion W := (W t ) t∈ [0,1] in the space of continuous functions C([0, 1]) as n tends to infinity. The first proof without nonstandard methods follows from the self-normalized Donsker theorem in [Csörgő et al. (2003)]. In this note, we present a very simple proof of the càdlàg version of this Poincaré-Borel lemma for Brownian motion. This is the content of Section 2. Some remarks on such Donsker-type convergence results on spheres with respect to ℓ p -norms are collected in Section 3. In fact, our simple approach can be used to to simplify the proof of the main result in [Csörgő et al. (2003)] as well. This is presented in Section 4.

Poincaré-Borel lemma for Brownian motion
Suppose X 1 , X 2 , . . . is an i.i.d. sequence of standand normal random variables. Then (X 1 , . . . , X n ) has a standard n-dimensional normal distribution. We define the processes with càdlàg paths Thus, Z n is equivalent to µ S,n • Q −1 for the functional and therefore a comparatively simple computation from the uniform distribution on the n-sphere. Then the following extension of the Poincaré-Borel lemma is true: Theorem 1. The sequence (Z n ) n∈N converges weakly in the Skorokhod space D([0, 1]) to a standard Brownian motion W as n tends to infinity.
Proof. The proof of the convergence of finite-dimensional distributions is in line with the classical Poincaré-Borel lemma: By the law of large numbers, 1 n n i=1 X 2 i → 1 in probability. Hence, by the continuous mapping theorem, √ n/( (X 1 , . . . , X n ) ) → 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 Z n and make use of a standard criterion. For all s ≤ t in [0, 1], we denote Due to the symmetry of the standard n-dimensional normal distributed vector (X 1 , . . . , X n ), for all pairwise different i, j, k, l, we observe We denote for shorthand m 1 := ⌊nt⌋ − ⌊nu⌋, m 2 := ⌊nu⌋ − ⌊ns⌋ and m 3 := n − (⌊nt⌋ − ⌊ns⌋). Then we observe for pairwise independent chi-squared random variables χ 2 m with m degrees of freedom. We recall Hence a computation via (2.3) yields Thus the well-known criterion ( [Billingsley (1968), Theorem 15.6], cp. the proof of [Sottinen (2001), Theorem 1]) implies the tightness of Z n .
Remark 2. (i) The heuristic connection of the Wiener measure and the uniform measure on an infinite-dimensional sphere goes back to Norbert Wieners study of the differential space, [Wiener (1923)]. The first informal presentation of Theorem 1 and further historical notes can be found in [McKean (1973)]. The first rigorous proof is given in Section 2 of [Cutland and Ng (1993)]. However, the authors make use of nonstandard analysis and the functional Q. To the best of our knowledge, the first proof of Theorem 1 follows from the general result [Csörgő et al. (2003), Theorem 1]. Our proof is a significant simplification, based on the pretty decoupling in the tightness argument. Moreover, this approach is extended in Section 4 to a simpler proof of [Csörgő et al. (2003) 3. ℓ n p -spheres In this section, we consider uniform measures on ℓ n p -spheres and prove that the limit in Theorem 1, i.e. p = 2, is the only case such that the simple pathwise functional Q leads to a nontrivial limit (Theorem 5). Furthermore, we present random variables living on ℓ n p -spheres, with a similar characterization for a fractional Brownian motion (Theorem 6). Concerning the ℓ n p norm x p = ( n i=1 |x i | p ) 1/p for p ∈ [1, ∞) and defining the ℓ n p unit sphere S n−1 p := {x ∈ R n : x p = 1}, the uniform measure µ S,n,p on S n−1 p is characterized similarly to the uniform measure on the euclidean unit sphere by independent results in [Schechtman and Zinn (1990), Lemma 1] and [Rachev and Rüschendorf (1991), Lemma 3.1]: Proposition 3. Suppose X, X 1 , X 2 , . . . is an i.i.d. sequence of random variables with density Then µ S,n,p d = (X 1 , . . . , X n ) (X 1 , . . . , X n ) p .
Remark 4. (i) We notice that the uniform measure on the ℓ n p -sphere equals the surface measure only in the cases p ∈ {1, 2, ∞}, see e.g. [Rachev and Rüschendorf (1991), Section 3] or the interesting study of the total variation distance of these measures for p ≥ 1 in [Naor and Romik (2003)].
(ii) In particular, we have a counterpart of the classical Poincaré-Borel lemma for finitedimensional distributions: For every fixed m ∈ N, n 1/p µ S,n,p • π −1 n,m converges in distribution to the random vector (X 1 , . . . , X m ) as n tends to infinity. This follows immediately from E[|X| p ] = 1 and the law of large numbers, cf. [Lifshits (2012), Proposition 6.1] or the finite-dimensional convergence in Theorem 1.
Similarly to the characterization of the central limit theorem, cp. [Kallenberg (2002), Theorem 4.23], but in contrast to the convergence of the projection on a finite number of coordinates in Remark 4, we have a uniqueness result for the processes constructed according to the pathwise functional Q. In the following we denote the convergence in distribution by d → and the almost sure convergence by a.s.
Remark 7. It would be interesting, whether the processes in Theorem 6 converging to the fractional Brownian motion could be constructed via measuresμ S,n,p on the ℓ n p -spheres and some simple pathwise functionals similar to Q p ?

The self-normalized Donsker theorem
Suppose X, X 1 , X 2 , . . . is an i.i.d. sequence of nondegenerate random variables and we denote for all n ∈ N, Limit theorems for self-normalized sums S n /V n play an important role in statistics, see e.g. [Giné et al. (1997)] and have been extensively studied during the last decades, cf. the monograph on self-normalizes processes [de la Peña et al. (2009)].
In [Csörgő et al. (2003)], the following invariance principle for self-normalized sum processes is established.
Thanks to a tightness argument as in the proof of Theorem 1, we obtain a simpler alternative for the proof.
Remark 10. (i) By the same reasoning, we obtain Theorem 5 for the i.i.d. sequence X, X 1 , X 2 , . . . such that Theorem 8 (a) is fulfilled.
(ii) In [Csörgő and Hu (2015)], a similar counterpart of Theorem 8 for α-stable Lévy processes is established. An interesting question would be on an uniqueness result similar to Theorem 5.