1 Main result

The focus of this paper is the volume of random projections of the cube \(B_{\infty }^N=[-1,1]^N\) in \(\mathbb R ^N\). To fix the notation, let \(n\geqslant 1\) be an integer and for \(N\geqslant n\), let \(G_{N,n}\) denote the Grassmannian manifold of all \(n\)-dimensional linear subspaces of \(\mathbb R ^N\). Equip \(G_{N,n}\) with the Haar probability measure \(\nu _{N,n}\), which is invariant under the action of the orthogonal group. Suppose that \((E(N))_{N\geqslant n}\) is a sequence of random subspaces with \(E(N)\) distributed according to \(\nu _{N,n}\). We consider the random variables

$$\begin{aligned} Z_N = |P_{E(N)} B_{\infty }^N|, \end{aligned}$$
(1.1)

where \(P_{E(N)}\) denotes the orthogonal projection onto \(E(N)\) and \(|\cdot |\) is \(n\)-dimensional volume, when \(n\) is fixed and \(N\rightarrow \infty \). We show that \(Z_N\) satisfies the following central limit theorem.

Theorem 1.1

$$\begin{aligned} \frac{Z_N-\mathbb{E }Z_N}{\sqrt{\mathop {\mathrm{var}}(Z_N)}}\overset{d}{\rightarrow } \mathcal N (0,1)\quad \text {as } N\rightarrow \infty . \end{aligned}$$
(1.2)

Here \(\overset{d}{\rightarrow }\) denotes convergence in distribution and \(\mathcal N (0,1)\) a standard Gaussian random variable with mean \(0\) and variance \(1\). Our choice of scaling for the cube is immaterial as the quantity in (1.2) is invariant under scaling and translation of \([-1,1]^N\).

Gaussian random matrices play a central role in the proof of Theorem 1.1, as is often the case with results about random projections onto subspaces \(E\in G_{N,n}\). Specifically, we let \(G\) be an \(n\times N\) random matrix with independent columns \(g_1,\ldots ,g_N\) distributed according to standard Gaussian measure \(\gamma _n\) on \(\mathbb R ^n\), i.e.,

$$\begin{aligned} d\gamma _n(x) = (2\pi )^{-n/2}e^{-||x||_2^2/2}dx. \end{aligned}$$

We view \(G\) as a linear operator from \(\mathbb R ^N\) to \(\mathbb R ^n\). If \(C\subset \mathbb R ^N\) is any convex body, then

$$\begin{aligned} |GC| = \det {(GG^*)}^{\frac{1}{2}} |P_E C|, \end{aligned}$$
(1.3)

where \(E= \mathop {\mathrm{Range}}(G^*)\) is distributed uniformly on \(G_{N,n}\). Moreover, \(\det {(GG^*)}^{1/2}\) and \(|P_E C|\) are independent. The latter fact underlies the Gaussian representation of intrinsic volumes, as proved by Tsirelson in [24] (see also [28]); it is also used in R. Vitale’s probabilistic derivation of the Steiner formula [27]. Passing between Gaussian vectors and random orthogonal projections is useful in a variety of contexts, e.g., [1, 5, 6, 8, 12, 14, 16, 18]. As we will show, however, it is a delicate matter to use (1.3) to prove limit theorems, especially with the normalization required in Theorem 1.1. Our path will involve analyzing asymptotic normality of \(|G B_{\infty }^N|\) before dealing with the quotient \(|GB_{\infty }^N|/\det {(GG^*)}^{1/2}\).

The set

$$\begin{aligned} GB_{\infty }^{N} = \left\{ \sum _{i=1}^N \lambda _i g_i:|\lambda _i|\leqslant 1, i=1,\ldots , N \right\} \end{aligned}$$

is a random zonotope, i.e., a Minkowski sum of the random segments \([-g_i,g_i]=\{\lambda g_i: |\lambda |\leqslant 1 \}\). By the well-known zonotope volume formula (e.g. [15]), \(X_N=|GB_{\infty }^N|\) satisfies

$$\begin{aligned} X_N = 2^n \sum _{1\leqslant i_1<\cdots <i_n\leqslant N}|\det {[g_{i_1}\cdots g_{i_n}]}|, \end{aligned}$$
(1.4)

where \(\det {[g_{i_1}\cdots g_{i_n}]}\) is the determinant of the matrix with columns \(g_{i_1},\ldots ,g_{i_n}\). The quantity

$$\begin{aligned} U_N = \frac{1}{{N\atopwithdelims ()n}} \sum _{1\leqslant i_1<\cdots <i_n\leqslant N}|\det {[g_{i_1}\cdots g_{i_n}]}| \end{aligned}$$

is a U-statistic and central limit theorems for U-statistics go back to Hoeffding [11]. In fact, formula (1.4) for \(X_N\) is simply a special case of Minkowski’s theorem on mixed volumes of convex sets (see §2). In [26], Vitale proved a central limit theorem for Minkowski sums of more general random convex sets, using mixed volumes and U-statistics (discussed in detail below). In particular, it follows from Vitale’s results that \(X_N\) satisfies a central limit theorem, namely,

$$\begin{aligned} \frac{X_N-\mathbb{E }X_N}{s_{N,n}} \overset{d}{\rightarrow } \mathcal N (0,1), \end{aligned}$$
(1.5)

where \(s_{N,n}\) is a certain conditional standard deviation (see Corollary 3.4). It follows that \(X_N\) also satisfies the central limit theorem with the canonical normalization:

$$\begin{aligned} \frac{X_N- \mathbb{E }X_N}{\sqrt{\mathop {\mathrm{var}}(X_N)}} \overset{d}{\rightarrow } \mathcal N (0,1) \quad \text {as } N\rightarrow \infty \end{aligned}$$
(1.6)

(using, e.g., Theorem 3.1 or Proposition 4.2).

It is tempting to think that the latter central limit theorem for \(X_N\) easily yields Theorem 1.1. However, for a family of convex bodies \(C=C_N\subset \mathbb R ^N, N=n, n+1, \ldots \), asymptotic normality of \(|GC|\) is not sufficient to conclude that \(|P_{E(N)} C|\) is asymptotically normal. For example, if \(C=B_2^N\) is the Euclidean ball in \(\mathbb R ^N\), then \(|GB_2^N|= \det {(GG^*)}^{1/2}|B_2^n|\) is asymptotically normal (e.g., [2, Theorems 4.2.3, 7.5.3]), however \(|P_{E(N)}B_2^N|\) is constant.

In fact, as we show in Proposition 4.4, both \(X_N\) and \(\det {(GG^*)}^{1/2}\) contribute to asymptotic normality of \(Z_N=|P_{E(N)}B_{\infty }^N|\), a technical difficulty that requires careful analysis. In particular, we invoke a randomization inequality from [7, Chapter 3] to deal with the canonical normalization for \(Z_N\) in Theorem 1.1. As a by-product, we also obtain the limiting behavior of the variance of \(Z_N\) as \(N\rightarrow \infty \).

We mention that when \(n=1\), Theorem 1.1 implies that if \((\theta _N)\) is a sequence of random vectors with \(\theta _N\) distributed uniformly on the sphere \(S^{N-1}\), then the \(\ell _1\)-norm \(||\cdot ||_1\) (the support function of the cube) satisfies

$$\begin{aligned} \frac{||\theta _N||_1- \mathbb{E }||\theta _N||_1}{\sqrt{\mathop {\mathrm{var}}(||\theta _N||_1)}} \overset{d}{\rightarrow } \mathcal N (0,1)\quad \text {as } N\rightarrow \infty . \end{aligned}$$

Theorem 1.1 complements recent research on central limit phenomena for various quantities that arise in Asymptotic Geometric Analysis; see, e.g., the papers of Bárány and Vu [4], Klartag [13], Reitzner [20] and the references therein. In fact, the central limit theorem for \(X_N\) in (1.6) can be seen as a counter-part to the Bárány-Vu result for convex hulls of Gaussian vectors [4]. Namely, when \(n\geqslant 2\) the quantity \(V_N = |\mathop {\mathrm{conv}}\left\{ g_1,\ldots ,g_N\right\} |\) satisfies

$$\begin{aligned} \frac{V_N- \mathbb{E }V_N}{\sqrt{\mathop {\mathrm{var}}(V_N)}}\overset{d}{\rightarrow } \mathcal{N }(0,1)\quad \text {as } N\rightarrow \infty ; \end{aligned}$$

see the latter article for the corresponding Berry-Esseen type estimate. The latter result is one of several recent deep central limit theorems in stochastic geometry concerning random convex hulls, e.g., [3, 29]. The techniques used in this paper are different and the main focus here is to understand the Grassmannian setting.

Lastly, for a thorough exposition of the properties of the cube, see [30].

2 Preliminaries

The setting is \(\mathbb R ^n\) with the usual inner-product \(\langle \cdot , \cdot \rangle \) and Euclidean norm \(||\cdot ||_2\); \(n\)-dimensional Lebesgue measure is denoted by \(|\cdot |\). For sets \(A,B \subset \mathbb R ^n\) and scalars \(\alpha , \beta \in \mathbb R \), we define \(\alpha A +\beta B\) by usual scalar multiplication and Minkowski addition: \(\alpha A +\beta B = \{\alpha a +\beta b: a\in A, b\in B\}\).

2.1 Mixed volumes

The mixed volume \(V(K_1,\ldots ,K_n)\) of compact convex sets \(K_1,\ldots ,K_n\) in \(\mathbb R ^n\) is defined by

$$\begin{aligned} V(K_1,\ldots ,K_n) = \frac{1}{n!} \sum _{j=1}^n (-1)^{n+j} \sum _{i_1<\cdots < i_j}\left|K_{i_1}+\cdots +K_{i_j}\right|. \end{aligned}$$

By a theorem of Minkowski, if \(t_1,\ldots ,t_N\) are non-negative real numbers then the volume of \(K=t_1K_1+\cdots +t_NK_N\) can be expressed as

$$\begin{aligned} \left|K\right| = \sum _{i_1=1}^N\cdots \sum _{i_n=1}^N V(K_{i_1},\ldots ,K_{i_n})t_{i_1}\cdots t_{i_n}. \end{aligned}$$
(2.1)

The coefficients \(V(K_{i_1},\ldots , K_{i_n})\) are non-negative and invariant under permutations of their arguments. When the \(K_i\)’s are origin-symmetric line segments, say \(K_i=[-x_i,x_i]=\{\lambda x_i:|\lambda |\leqslant 1\}\), for some \(x_1,\ldots ,x_n\in \mathbb R ^n\), we simplify the notation and write

$$\begin{aligned} V(x_1,\ldots ,x_n)=V([-x_1,x_1],\ldots ,[-x_n,x_n]). \end{aligned}$$
(2.2)

We will make use of the following properties:

  1. (i)

    \(V(K_1,\ldots ,K_n)>0\) if and only if there are line segments \(L_i\subset K_i\) with linearly independent directions.

  2. (ii)

    If \(x_1,\ldots , x_n\in \mathbb R ^n\), then

    $$\begin{aligned} n!V(x_1,\ldots ,x_n) = 2^n|\det {[x_1\cdots x_n]}|, \end{aligned}$$
    (2.3)

    where \(\det {[x_1\cdots x_n]}\) denotes the determinant of the matrix with columns \(x_1,\ldots ,x_n\).

  3. (iii)

    \(V(K_1,\ldots ,K_n)\) is increasing in each argument (with respect to inclusion).

For further background we refer the reader to [22, Chapter 5] or [10], Appendix A].

A zonotope is a Minkowski sum of line segments. If \(x_1,\ldots ,x_N\) are vectors in \(\mathbb R ^n\), then

$$\begin{aligned} \sum _{i=1}^N[-x_i,x_i] =\left\{ \sum _{i=1}^N\lambda _i x_i: |\lambda _i|\leqslant 1, \;i=1,\ldots ,N\right\} . \end{aligned}$$

Alternatively, a zonotope can be seen as a linear image of the cube \(B_{\infty }^N = [-1,1]^N\). If \(x_1,\ldots ,x_N\in \mathbb R ^n\), one can view the \(n\times N\) matrix \(X = [x_1\cdots x_N]\) as a linear operator from \(\mathbb R ^N\) to \(\mathbb R ^n\); in this case, \(X B_{\infty }^N = \sum _{i=1}^N [-x_i,x_i]\).

By (2.1) and properties (i) and (ii) of mixed volumes, the volume of \(\sum _{i=1}^N[-x_i,x_i]\) satisfies

$$\begin{aligned} \Bigl |\sum _{i=1}^N[-x_i,x_i]\Bigr |= 2^n \sum _{1\leqslant i_1<\cdots <i_n\leqslant N} |\det {[x_{i_1}\cdots x_{i_n}]}|. \end{aligned}$$
(2.4)

Note that for \(x_1,\ldots ,x_n\in \mathbb R ^n\),

$$\begin{aligned} |\det {[x_1\cdots x_n]}| = ||x_1||_2||P_{F_1^{\perp }}x_2||_2 \cdots ||P_{F_{n-1}^{\perp }}x_n ||_2, \end{aligned}$$
(2.5)

where \(F_k=\mathop {\mathrm{span}}\{x_1,\ldots ,x_k\}\) for \(k=1,\ldots ,n-1\) (which can be proved using Gram-Schmidt orthogonalization, e.g., [2, Theorem 7.5.1]).

We will also use the Cauchy-Binet formula. Let \(x_1,\ldots , x_N\in \mathbb R ^n\) and let \(X\) be the \(n\times N\) matrix with columns \(x_1,\ldots ,x_N\), i.e., \(X= [x_1\cdots x_N]\). Then

$$\begin{aligned} \det {(XX^*)}^{\frac{1}{2}} = \left( \sum _{1\leqslant i_1<\cdots <i_n\leqslant N} \det {[x_{i_1}\cdots x_{i_n}]}^2\right) ^{\frac{1}{2}}; \end{aligned}$$
(2.6)

for a proof, see e.g. [9, §3.2].

2.2 Slutsky’s theorem

We will make frequent use of Slutsky’s theorem on convergence of random variables (see, e.g., [23, §1.5.4]).

Theorem 2.1

Let \((X_N)\) and \((\alpha _N)\) be sequences of random variables. Suppose that \(X_N\overset{d}{\rightarrow } X_0\) and \(\alpha _N\overset{\mathbb{P }}{\rightarrow } \alpha _0\), where \(\alpha _0\) is a finite constant. Then

$$\begin{aligned} X_N + \alpha _N \overset{d}{\rightarrow } X_0 + \alpha _0 \end{aligned}$$

and

$$\begin{aligned} \alpha _N X_N \overset{d}{\rightarrow } \alpha _0 X_0. \end{aligned}$$

Slutsky’s theorem also applies when the \(X_N\)’s take values in \(\mathbb R ^k\) and satisfy \(X_N\overset{d}{\rightarrow } X_0\) and \((A_N)\) is a sequence of \(m\times k\) random matrices such that \(A_N\overset{\mathbb{P }}{\rightarrow } A_0\) and the entries of \(A_0\) are constants. In this case, \(A_N X_N\overset{d}{\rightarrow } A_0 X_0\).

3 U-statistics

In this section, we give the requisite results from the theory of U-statistics needed to prove asymptotic normality of \(X_N\) and \(Z_N\) stated in the introduction. For further background on U-statistics, see e.g. [7, 21, 23].

Let \(X_1,X_2,\ldots \) be a sequence of i.i.d. random variables with values in a measurable space \((S, \mathcal{S })\). Let \(h:S^m \rightarrow \mathbb R \) be a measurable function. For \(N\geqslant m\), the U-statistic of order \(m\) with kernel \(h\) is defined by

$$\begin{aligned} U_N = U_N(h) = \frac{(N-m)!}{N!} \sum _{(i_1,\ldots ,i_m)\in I_N^m} h(X_{i_1},\ldots ,X_{i_m}), \end{aligned}$$
(3.1)

where

$$\begin{aligned} I_N^m=\left\{ (i_1,\ldots ,i_m):i_j\in \mathbb N , 1\leqslant i_j\leqslant N, i_j\not = i_k \text { if } j\not =k \right\} . \end{aligned}$$

When \(h\) is symmetric, i.e., \(h(x_1, \ldots , x_m) = h(x_{\sigma (1)}, \ldots , x_{\sigma (m)})\) for every permutation \(\sigma \) of \(m\) elements, we can write

$$\begin{aligned} U_N = U(X_1, \ldots ,X_N) = \frac{1}{ {N \atopwithdelims ()m}} \sum _{1\leqslant i_1<\cdots <i_m\leqslant N} h( X_{i_1}, \ldots , X_{i_m}); \end{aligned}$$
(3.2)

here the sum is taken over all \({N \atopwithdelims ()m}\) subsets \(\{i_1,\ldots , i_m\}\) of \(\{1,\ldots , N\}\).

Using the latter notation, we state several well-known results, due to Hoeffding (see, e.g., [23, Chapter 5]).

Theorem 3.1

For \(N\geqslant m\), let \(U_N\) be a U-statistic with kernel \(h:S^m \rightarrow \mathbb R \). Set \(\zeta = \mathop {\mathrm{var}}(\mathbb{E }[h(X_1,\ldots ,X_m) | X_1])\).

  1. (1)

    The variance of \(U_N\) satisfies

    $$\begin{aligned} \mathop {\mathrm{var}}(U_N) = \frac{m^2 \zeta }{N} + O(N^{-2})\quad \text {as }N\rightarrow \infty . \end{aligned}$$
  2. (2)

    If  \(\mathbb{E }|h(X_1,\ldots ,X_m)| <\infty \), then \(U_N \overset{a.s.}{\rightarrow } \mathbb{E }U_N\) as \(N\rightarrow \infty \).

  3. (3)

    If  \(\mathbb{E }h^2(X_1, \ldots , X_m)< \infty \) and \(\zeta >0\), then

    $$\begin{aligned} \sqrt{N} \left( \frac{ U_{N}- \mathbb{E }U_N}{ m\sqrt{\zeta } }\right) \overset{d}{\rightarrow } \mathcal{{N}}(0,1)\quad \text {as } N\rightarrow \infty . \end{aligned}$$

The corresponding Berry-Esseen type bounds are also available (see, e.g,. [23, page 193]), stated here in terms of the function

$$\begin{aligned} \Phi (t)=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^t e^{-s^2/2}ds. \end{aligned}$$

Theorem 3.2

With the preceding notation, suppose that \(\xi =\mathbb{E }|h(X_{1}, \ldots , X_{m})|^3 <\infty \) and

$$\begin{aligned} \zeta = \mathop {\mathrm{var}}(\mathbb{E }[ h(X_1,\ldots ,X_m) | X_1])>0. \end{aligned}$$

Then

$$\begin{aligned} \sup _{t\in \mathbb R } \left|\mathbb P \left( \sqrt{N}\left( \frac{U_N-\mathbb{E }U_N}{m\sqrt{\zeta }} \right) \leqslant t\right) - \Phi (t)\right|\leqslant \frac{c \xi }{ (m^{2} \zeta )^{\frac{3}{2}}\sqrt{N}}, \end{aligned}$$

where \(c>0\) is an universal constant.

3.1 U-statistics and mixed volumes

Let \(\mathcal C _n\) denote the class of all compact, convex sets in \(\mathbb R ^n\). A topology on \(\mathcal C _n\) is induced by the Hausdorff metric

$$\begin{aligned} \delta ^H(K,L) = \inf \{\delta >0: K \subset L +\delta B_2^n, L\subset K+\delta B_2^n\}. \end{aligned}$$

A random convex set is a Borel measurable map from a probability space into \(\mathcal{C }_n\). A key ingredient in our proof is the following theorem for Minkowski sums of random convex sets due to Vitale [26]; we include the proof for completeness.

Theorem 3.3

Let \(n\geqslant 1\) be an integer. Suppose that \(K_1,K_2,\ldots \) are i.i.d. random convex sets in \(\mathbb R ^n\) such that \(\mathbb{E }\sup _{x\in K_1}||x||_2<\infty \). Set \(V_N =|\sum _{i=1}^N K_i|\) and suppose that \(\mathbb{E }V(K_1,\ldots ,K_n)^2<\infty \) and furthermore that \(\zeta = \mathop {\mathrm{var}}(\mathbb{E }[V(K_1,\ldots ,K_n)| K_1]) >0\). Then

$$\begin{aligned} \sqrt{N}\left( \frac{V_N - \mathbb{E }V_N}{ (N)_n n\sqrt{\zeta }}\right) \overset{d}{\rightarrow } \mathcal N (0,1) \quad \text {as } N\rightarrow \infty , \end{aligned}$$

where \((N)_n=\frac{N!}{(N-n)!}\).

Proof

Taking \(h:(\mathcal C _n)^n \rightarrow \mathbb R \) to be \(h(K_1,\ldots ,K_n) = V(K_1,\ldots ,K_n)\) and using (2.1), we have

$$\begin{aligned} \frac{1}{(N)_n}V_N = U_N + \frac{1}{(N)_n}\sum _{(i_1,\ldots ,i_n)\in J}V(K_{i_1},\ldots ,K_{i_n}) \end{aligned}$$
(3.3)

where

$$\begin{aligned} U_N = \frac{1}{(N)_n} \sum _{(i_1,\ldots ,i_n)\in I_N^n} V(K_{i_1},\ldots ,K_{i_n}), \end{aligned}$$

and \(J=\{1,\ldots ,N\}^n\backslash I_N^n\). Note that \(|J|/(N)_n = O(\frac{1}{N})\) and thus the second term on the right-hand side of (3.3) tends to zero in probability. Applying Theorem 3.1(3) and Slutsky’s theorem leads to the desired conclusion.\(\square \)

In the special case when the \(K_i\)’s are line segments, say \(K_i = [-X_i, X_i]\) where \(X_1,X_2,\ldots \) are i.i.d. random vectors in \(\mathbb R ^n\), the assumptions in the latter theorem can be readily verified by using (2.3). Furthermore, if the \(X_i\)’s are rotationally-invariant, the assumptions simplify further as follows (essentially from [26], stated here in a form that best serves our purpose).

Corollary 3.4

Let \(X=R\theta \) be a random vector such that \(\theta \) is uniformly distributed on the sphere \(S^{n-1}\) and \(R\geqslant 0\) is independent of \(\theta \) and satisfies \(\mathbb{E }R^2 <\infty \) and \(\mathop {\mathrm{var}}(R)>0\). For each \(i=1,2,\ldots \), let \(X_i=R_i\theta _i\) be independent copies of \(X\). Let \(D_n=|\det {[\theta _1\cdots \theta _n]}|\) and set

$$\begin{aligned} \zeta _1 = 4^n \mathop {\mathrm{var}}(R) \mathbb{E }^{2(n-1)} R \mathbb{E }^2 D_n. \end{aligned}$$

Then \(V_N=|\sum _{i=1}^N [-X_i,X_i]|\) satisfies

$$\begin{aligned} \sqrt{N}\left( \frac{V_N - \mathbb{E }V_N}{{N\atopwithdelims ()n} n\sqrt{\zeta _1}}\right) \rightarrow \mathcal N (0,1)\quad \text {as } N\rightarrow \infty . \end{aligned}$$

Proof

Plugging \(X_i=R_i\theta _i, i=1,\ldots ,n\), into (2.3) gives

$$\begin{aligned} n! V(X_1,\ldots ,X_n) = 2^n R_1\cdots R_n D_n. \end{aligned}$$
(3.4)

By (2.5),

$$\begin{aligned} D_n = ||\theta _1||_2||P_{{F_1}^{\perp }}\theta _2||_2\cdots ||P_{F_{n-1}^{\perp }}\theta _n||_2, \end{aligned}$$
(3.5)

with \(F_k = \mathop {\mathrm{span}}\{\theta _1,\ldots ,\theta _k\}\) for \(k=1,\ldots ,n-1\). In particular, \(D_n\leqslant 1\) and thus (3.4) implies

$$\begin{aligned} \mathbb{E }V(X_1,\ldots ,X_n)^2 \leqslant \frac{4^n}{(n!)^2}\mathbb{E }^n R^2 <\infty . \end{aligned}$$

Using (3.4) once more, together with (3.5), we have

$$\begin{aligned} n!\mathbb{E }[ V(X_1,\ldots ,X_n)\vert X_1 ] = 2^n R_1 \mathbb{E }R_2\cdots \mathbb{E }R_n \mathbb{E }D_n; \end{aligned}$$
(3.6)

here we have used the fact that \(\mathbb{E }||P_{{F_k}^{\perp }}\theta _{k+1}||_2\) depends only on the dimension of \(F_k\) (which is equal to \(k\) a.s.) and that \(||\theta _1||_2=1\) a.s. By (3.6) and our assumption \(\mathop {\mathrm{var}}(R)>0\), we can apply Theorem 3.3 with

$$\begin{aligned} \zeta = \mathop {\mathrm{var}}(\mathbb{E }[V(X_1,\ldots ,X_n)| X_1]) = \frac{\zeta _1}{(n!)^2} >0, \end{aligned}$$

where \(\zeta _1\) is defined in the statement of the corollary.\(\square \)

For further information on Theorem 3.3, including a CLT for the random sets themselves, or the case when \(\zeta =0\), see [26] or [17, p. 232]; see also [25].

Corollary 3.4 implies the first central limit theorem for \(X_N\) stated in the introduction (1.5). To prove Theorem 1.1, however, we will need some additional tools.

3.2 Randomization

In this subsection, we discuss a randomization inequality for U-statistics. It will be used for variance estimates and will play a crucial role in the proof of Theorem 1.1.

Using the notation at the beginning of §3, suppose that \(h:(\mathbb R ^n)^m\rightarrow \mathbb R \) satisfies \(\mathbb{E }|h(X_1,\ldots ,X_m)| <\infty \) and let \(1<r\leqslant m\). Following [7, Definition 3.5.1], we say that \(h\) is degenerate of order \(r-1\) if

$$\begin{aligned} \mathbb{E }_{X_r,\ldots ,X_m}h(x_1,\ldots ,x_{r-1},X_r,\ldots ,X_m) = \mathbb{E }h(X_1,\ldots , X_m) \end{aligned}$$

for all \(x_1,\ldots ,x_{r-1}\in \mathbb R ^n\), and the function

$$\begin{aligned} S^r \ni (x_1,\ldots ,x_r)\mapsto \mathbb{E }_{X_{r+1},\ldots ,X_m}h(x_1,\ldots ,x_r,X_{r+1},\ldots ,X_m) \end{aligned}$$

is non-constant. If \(h\) is not degenerate of any positive order \(r\), we say it is non-degenerate or degenerate of order \(0\). We will make use of the following randomization theorem, which is a special case of [7, Theorem 3.5.3].

Theorem 3.5

Let \(1\leqslant r \leqslant m\) and \(p\geqslant 1\). Suppose that \(h:S^m\rightarrow \mathbb R \) is degenerate of order \(r-1\) and \(\mathbb{E }|h(X_1,\ldots ,X_m)|^p<\infty \). Set

$$\begin{aligned} f(x_1,\ldots ,x_m) = h(x_1,\ldots ,x_m)-\mathbb{E }h(X_1,\ldots ,X_m). \end{aligned}$$

Let \(\varepsilon _1,\ldots ,\varepsilon _N\) denote i.i.d. Rademacher random variables, independent of \(X_1,\ldots ,X_N\). Then

$$\begin{aligned}&\mathbb{E }\bigl \vert \sum _{(i_1,\ldots ,i_m)\in I_N^m} f(X_{i_1},\ldots ,X_{i_m}) \bigr \vert ^p\\&\quad \simeq _{m,p} \mathbb{E }\bigl |\sum _{(i_1,\ldots ,i_m)\in I_N^m} \varepsilon _{i_1}\cdots \varepsilon _{i_r} f(X_{i_1},\ldots , X_{i_m})\bigr |^p. \end{aligned}$$

Here \(A\simeq _{m,p}B\) means \(C^{\prime }_{m,p} A \leqslant B \leqslant C^{\prime \prime }_{m,p} A\), where \(C^{\prime }_{m,p}\) and \(C^{\prime \prime }_{m,p}\) are constants that depend only on \(m\) and \(p\).

Corollary 3.6

Let \(\mu \) be a probability measure on \(\mathbb R ^n\), absolutely continuous with respect to Lebesgue measure. Suppose that \(X_1,\ldots ,X_N\) are i.i.d. random vectors distributed according to \(\mu \). Let \(p\geqslant 2\) and suppose \(\mathbb{E }|\det {[X_1\cdots X_n]}|^p<\infty \). Define \(f:(\mathbb R ^n)^n \rightarrow \mathbb R \) by

$$\begin{aligned} f(x_1,\ldots ,x_n) = |\det {[x_1\cdots x_n]}| - \mathbb{E }|\det { [X_1\cdots X_n]}|. \end{aligned}$$

Then

$$\begin{aligned} \mathbb{E }\bigl |\sum _{1\leqslant i_1<\ldots <i_n\leqslant N} f(X_{i_1},\ldots ,X_{i_n}) \bigr |^p \leqslant C_{n,p} N^{p(n -\frac{1}{2})} \mathbb{E }|f(X_1,\ldots ,X_n)|^p, \end{aligned}$$

where \(C_{n,p}\) is a constant that depends on \(n\) and \(p\).

Proof

Since \(\mu \) is absolutely continuous, \(\mathop {\mathrm{dim}}\left( \mathop {\mathrm{span}}\{X_1,\ldots , X_k\}\right) =k\) a.s. for \(k=1,\ldots , n\). Moreover, \( f(ax_1,\ldots , x_n) = |a|f(x_1,\ldots ,x_n)\) for any \(a\in \mathbb R \), hence \(f\) is non-degenerate [(cf. (2.5)]. Thus we may apply Theorem 3.5 with \(r=1\):

$$\begin{aligned} \mathbb{E }\Bigl |\sum _{1\leqslant i_1<\cdots <i_n\leqslant N} n! f(X_{i_1},\ldots ,X_{i_n}) \Bigr |^p&= \mathbb{E }\Bigl |\sum _{(i_1,\ldots ,i_n)\in I_N^n} f(X_{i_1},\ldots ,X_{i_n}) \Bigr |^p\\&\leqslant C_{n,p} \mathbb{E }\Bigl |\sum _{(i_1,\ldots ,i_n)\in I_N^n} \varepsilon _{i_1} f(X_{i_1},\ldots ,X_{i_n}) \Bigr |^p. \end{aligned}$$

Suppose now that \(X_1,\ldots ,X_N\) are fixed. Taking expectation in \(\mathbf{\varepsilon }= (\varepsilon _1,\ldots ,\varepsilon _N)\) and appling Khintchine’s inequality and then Hölder’s inequality twice, we have

$$\begin{aligned}&\mathbb{E }_\mathbf{\varepsilon } \Bigl |\sum _{(i_1,\ldots ,i_n)\in I_N^n} \varepsilon _{i_1} f(X_{i_1},\ldots ,X_{i_n}) \Bigr |^p\\&\quad = \mathbb{E }_\mathbf{\varepsilon }\Bigl |\sum _{i_1=1}^N \varepsilon _{i_1} \sum _{\begin{array}{c} (i_2,\ldots ,i_n)\\ (i_1,\ldots ,i_n)\in I_N^n \end{array}} f(X_{i_1},\ldots ,X_{i_n})\Bigr |^p\\&\quad \leqslant C\Bigl |\sum _{i_1=1}^N \Bigl ( \sum _{\begin{array}{c} (i_2,\ldots ,i_n)\\ (i_1,\ldots ,i_n)\in I_N^n \end{array}} f(X_{i_1},\ldots ,X_{i_n})\Bigr )^2\Bigr |^{\frac{p}{2}}\\&\quad \leqslant C\left( {N-1 \atopwithdelims ()n-1} (n-1)!\right) ^{\frac{p}{2}} \Bigl |\sum _{(i_1,\ldots ,i_n)\in I_N^n} f(X_{i_1},\ldots ,X_{i_n})^2\Bigr |^{\frac{p}{2}} \\&\quad \leqslant C\left( {N-1 \atopwithdelims ()n-1} (n-1)!\right) ^{\frac{p}{2}} \left( {N \atopwithdelims ()n }n!\right) ^{\frac{p-2}{2}} \sum _{(i_1,\ldots ,i_n)\in I_N^n} |f(X_{i_1},\ldots ,X_{i_n})|^p, \end{aligned}$$

where \(C\) is an absolute constant. Taking expectation in the \(X_i\)’s gives

$$\begin{aligned}&\mathbb{E }\Bigl |\sum _{(i_1,\ldots ,i_n)\in I_N^n} \varepsilon _{i_1} f(X_{i_1},\ldots ,X_{i_n}) \Bigr |^p\\&\quad \leqslant \left( {N-1 \atopwithdelims ()n-1} (n-1)!\right) ^{\frac{p}{2}} \left( {N \atopwithdelims ()n }n!\right) ^{\frac{p-2}{2}} {N\atopwithdelims ()n}n! \mathbb{E }|f(X_{1},\ldots ,X_{n})|^p. \end{aligned}$$

The proposition follows as stated by using the estimate \({N\atopwithdelims ()n}\leqslant (eN/n)^n\).\(\square \)

4 Proof of Theorem 1.1

As explained in the introduction, our first step is identity (1.3), the proof of which is included for completeness.

Proposition 4.1

Let \(N\geqslant n\) and let \(G\) be an \(n\times N\) random matrix with i.i.d. standard Gaussian entries. Let \(C\subset \mathbb R ^N\) be a convex body. Then

$$\begin{aligned} \left|GC\right| = \det {(GG^*)}^{\frac{1}{2}} \left|P_E C\right|, \end{aligned}$$
(4.1)

where \(E = \mathop {{\mathrm{Range}}(G^*)}\). Moreover, \(E\) is distributed uniformly on \(G_{N,n}\) and \(\det {(GG^*)}^{\frac{1}{2}}\) and \(\left|P_E C\right|\) are independent.

Proof

Identity (4.1) follows from polar decomposition; see, e.g., [18, Theorem 2.1(iii)]. To prove that the two factors are independent, we note that if \(U\) is an orthogonal transformation, we have \(\mathop {\mathrm{det}}(GG^*)^{1/2} = \mathop {\mathrm{det}}((GU)(GU)^*)^{1/2}\); moreover, \(G\) and \(GU\) have the same distribution. Thus if \(U\) is a random orthogonal transformation distributed according to the Haar measure, we have for \(s, t\geqslant 0\),

$$\begin{aligned}&{\mathbb{P }}_{\otimes \gamma _n}\left( \mathop {\mathrm{det}}(GG^*)^{1/2}\leqslant s, |P_{\mathop {\mathrm{Range}}(G^*)}C|\leqslant t\right) \\&\quad = {\mathbb{P }}_{\otimes \gamma _n}\otimes {\mathbb{P }}_{U}\left( \mathop {\mathrm{det}}(GG^*)^{1/2}\leqslant s, |P_{\mathop {\mathrm{Range}}(U^*G^*)}C|\leqslant t\right) \\&\quad =\mathbb{E }_{\otimes \gamma _n}\left( {{1\!\!1}}_{\{ \mathop {\mathrm{det}}(GG^*)^{1/2}\leqslant s\}} \mathbb{E }_U {{1\!\!1}}_{\{|P_{U^* \mathop {\mathrm{Range}}(G^*)} C|\leqslant t\}}\right) \\&\quad = {\mathbb{P }}_{\otimes \gamma _n}\left( \mathop {\mathrm{det}}(GG^*)^{1/2}\leqslant s\right) \nu _{N,n}\left( E\in G_{N,n}:|P_{E}C|\leqslant t\right) . \end{aligned}$$

\(\square \)

Taking \(C=B_{\infty }^N\) in (4.1), we set

$$\begin{aligned} X_N=\left|GB_{\infty }^N\right| = 2^n \sum _{1\leqslant i_1<\cdots <i_n\leqslant N}|\det {[g_{i_1}\cdots g_{i_n}]}| \end{aligned}$$
(4.2)

[cf. (2.4)],

$$\begin{aligned} Y_N = \det {(GG^*)}^{\frac{1}{2}} = \left( \sum _{1\leqslant i_1<\cdots < i_n \leqslant N} \det {[g_{i_1}\cdots g_{i_m}]}^2\right) ^{\frac{1}{2}} \end{aligned}$$
(4.3)

[cf. (2.6)], and

$$\begin{aligned} Z_N = \left|P_E B_{\infty }^N\right|, \end{aligned}$$
(4.4)

where \(E\) is distributed according to \(\nu _{N,n}\) on \(G_{N,n}\). Then \(X_N = Y_N Z_N\), where \(Y_N\) and \(Z_N\) are independent. In order to prove Theorem 1.1, we start with several properties of \(X_N\) and \(Y_N\).

Proposition 4.2

Let \(X_N\) be as defined in (4.2).

  1. (1)

    For each \(p\geqslant 2\),

    $$\begin{aligned} \mathbb{E }|X_N-\mathbb{E }X_N|^p\leqslant C_{n,p} N^{p(n-\frac{1}{2})}. \end{aligned}$$
  2. (2)

    The variance of \(X_N\) satisfies

    $$\begin{aligned} \frac{\mathop {\mathrm{var}}(X_N)}{N^{2n-1}} \rightarrow c_n \quad \text {as }N\rightarrow \infty , \end{aligned}$$

    where \(c_n\) is a positive constant that depends only on \(n\).

  3. (3)

    \(X_N\) is asymptotically normal; i.e.,

    $$\begin{aligned} \frac{X_N-\mathbb{E }X_N}{\sqrt{\mathop {\mathrm{var}}(X_N)}} \overset{d}{\rightarrow } \mathcal N (0,1) \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Proof

Statement (1) follows from Corollary 3.6.

To prove (2), let \(g\) be a random vector distributed according to \(\gamma _n\). Then Corollary 3.4 with \(\zeta _1 =4^n\mathop {\mathrm{var}}(||g||_2)\mathbb{E }^{2(n-1)}||g||_2\mathbb{E }^2 D_n \) yields

$$\begin{aligned} \sqrt{N}\left( \frac{X_N- \mathbb{E }X_N}{{N\atopwithdelims ()n}n\sqrt{\zeta _1}}\right) \overset{d}{\rightarrow }\mathcal N (0,1) \quad \text {as }N\rightarrow \infty . \end{aligned}$$
(4.5)

On the other hand, by part (1) we have

$$\begin{aligned} \frac{\mathbb{E }|X_N-\mathbb{E }X_N|^4}{N^{4n-2}}\leqslant C_{n,4}. \end{aligned}$$

This implies that the sequence \((X_N-\mathbb{E }X_N)/N^{n-\frac{1}{2}}\) is uniformly integrable, hence

$$\begin{aligned} \frac{\sqrt{\mathop {\mathrm{var}}(X_N)}}{N^{-\frac{1}{2}}{N\atopwithdelims ()n}n\sqrt{\zeta _1}} \rightarrow 1 \quad \text {as }N\rightarrow \infty , \end{aligned}$$

which implies (2).

Part (3) follows from (4.5) and Slutsky’s theorem.\(\square \)

We now turn to \(Y_N=\det {(GG^*)}^{\frac{1}{2}}\). It is well-known that

$$\begin{aligned} Y_N = \chi _N\chi _{N-1}\cdot \ldots \cdot \chi _{N-n+1}, \end{aligned}$$
(4.6)

where \(\chi _k=\sqrt{\chi _k^2}\) and the \(\chi _k^2\)’s are independent chi-squared random variables with \(k\) degrees of freedom, \(k=N, \ldots ,N-n+1\) (see, e.g., [2, Chapter 7]). Consequently,

$$\begin{aligned} \mathbb{E }Y_{N}^2 =\frac{N!}{(N-n)!} = N^n\left( 1-\frac{1}{N}\right) \cdots \left( 1-\frac{n-1}{N}\right) . \end{aligned}$$
(4.7)

Additionally, we will use the following basic properties of \(Y_N\).

Proposition 4.3

Let \(Y_N\) be as defined in (4.3).

  1. (1)

    For each \(p\geqslant 2\),

    $$\begin{aligned} \mathbb{E }|Y_N^2 - \mathbb{E }Y_N^2|^p \leqslant C_{n,p} N^{p(n-\frac{1}{2})}. \end{aligned}$$
  2. (2)

    The variance of \(Y_N\) satisfies

    $$\begin{aligned} \frac{\mathop {\mathrm{var}}(Y_{N})}{N^{n-1}} \rightarrow \frac{n}{2} \quad \text {as } N\rightarrow \infty . \end{aligned}$$
  3. (3)

    \(Y_N^2\) is asymptotically normal; i.e.,

    $$\begin{aligned} \sqrt{N}\left( \frac{Y_N^2}{N^n}-1\right) \overset{d}{\rightarrow } \mathcal N (0,2n) \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Proof

To prove part (1), we apply Corollary 3.6 to \(Y_N^2\).

To prove part (2), we use (4.6) and define \(Y_{N,n}\) by \(Y_{N,n}=Y_N = \chi _N\chi _{N-1}\cdot \ldots \cdot \chi _{N-n+1}\) and procede by induction on \(n\). Suppose first that \(n=1\) so that \(Y_{N,1} =\chi _N\). By the concentration of Gaussian measure (e.g., [19, Remark 4.8]), there is an absolute constant \(c_1\) such that \(\mathbb{E }|\chi _N - \mathbb{E }\chi _N|^4<c_1\) for all \(N\), which implies that the sequence \((\chi _N- \mathbb{E }\chi _N)_N\) is uniformly integrable. By the law of large numbers \(\chi _N/\sqrt{N}\rightarrow 1\) a.s. and hence \(\mathbb{E }\chi _N /\sqrt{N}\rightarrow 1\), by uniform integrability. Note that

$$\begin{aligned} \chi _N - \mathbb{E }\chi _N&= \frac{\chi _N^2 - \mathbb{E }^2 \chi _N}{\chi _N + \mathbb{E }\chi _N} \\&= \frac{\sqrt{N}}{\chi _N +\mathbb{E }\chi _N} \frac{\chi _N^2-N}{\sqrt{N}} +\frac{\sqrt{N}}{\chi _N+\mathbb{E }\chi _N}\frac{N-\mathbb{E }^2\chi _N}{\sqrt{N}}. \end{aligned}$$

By Slutsky’s theorem and the classical central limit theorem,

$$\begin{aligned} \frac{\sqrt{N}}{\chi _N +\mathbb{E }\chi _N} \frac{\chi _N^2-N}{\sqrt{N}} \overset{d}{\rightarrow }\frac{1}{2}\mathcal N (0,2)\quad \text {as } N\rightarrow \infty , \end{aligned}$$

while

$$\begin{aligned} \frac{\sqrt{N}}{\chi _N+\mathbb{E }\chi _N}\frac{N-\mathbb{E }^2\chi _N}{\sqrt{N}} \rightarrow 0\text { (a.s.)} \quad \text {as } N\rightarrow \infty , \end{aligned}$$

since \(\mathop {\mathrm{var}}(\chi _N)=N-\mathbb{E }^2 \chi _N <c_1^{1/2}\). Thus

$$\begin{aligned} \chi _N - \mathbb{E }\chi _N \overset{d}{\rightarrow } \frac{1}{2}\mathcal N (0,2) = \mathcal N (0,\frac{1}{2}) \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Appealing again to uniform integrability of \((\chi _N-\mathbb{E }\chi _N)_N\), we have

$$\begin{aligned} \mathop {\mathrm{var}}(Y_{N,1})= \mathbb{E }|\chi _N- \mathbb{E }\chi _N|^2 \rightarrow \frac{1}{2} \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Assume now that

$$\begin{aligned} \frac{\mathop {\mathrm{var}}(Y_{N-1,n-1})}{N^{n-2}} \rightarrow \frac{n-1}{2} \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Note that

$$\begin{aligned} \mathop {\mathrm{var}}(Y_{N,n})&= \mathbb{E }\chi _N^2 \mathbb{E }Y_{N-1,n-1}^2 - \mathbb{E }^2 \chi _N \mathbb{E }^2 Y_{N-1,n-1} \\&= \mathbb{E }(\chi _N^2 - \mathbb{E }^2 \chi _N) \mathbb{E }Y_{N-1,n-1}^2 + \mathbb{E }^2 \chi _N (\mathbb{E }Y_{N-1,n-1}^2 - \mathbb{E }^2 Y_{N-1,n-1}) \\&= \mathop {\mathrm{var}}(\chi _N) \mathbb{E }Y_{N-1,n-1}^2+ \mathbb{E }^2 \chi _N \mathop {\mathrm{var}}(Y_{N-1,n-1}). \end{aligned}$$

We conclude the proof of part (2) by applying (4.7),

$$\begin{aligned} \frac{\mathop {\mathrm{var}}(\chi _N)\mathbb{E }Y_{N-1,n-1}^2}{N^{n-1}} \rightarrow \frac{1}{2}, \end{aligned}$$

and, using the inductive hypothesis,

$$\begin{aligned} \frac{\mathbb{E }^2 \chi _N\mathop {\mathrm{var}}(Y_{N-1,n-1})}{N^{n-1}}\rightarrow \frac{n-1}{2}. \end{aligned}$$

Lastly, statement (3) is well-known (see, e.g., [2, §7.5.3]).\(\square \)

The next proposition is the key identity for \(Z_N\). To state it we will use the following notation:

$$\begin{aligned} \Delta _{n,p}^p= \mathbb{E }|\mathop {\mathrm{det}}[g_1\cdots g_n]|^p. \end{aligned}$$
(4.8)

Explicit formulas for \(\Delta _{n,p}^p\) are well-known and follow from identity (2.5); see, e.g., [2, p. 269].

Proposition 4.4

Let \(X_N, Y_N\) and \(Z_N\) be as above (cf. (4.2)–(4.4)). Then

$$\begin{aligned} \frac{Z_N-\mathbb{E }Z_N}{N^{\frac{n-1}{2}}} = \alpha _{N,n} \frac{X_N-\mathbb{E }X_N}{N^{n-\frac{1}{2}}} -\beta _{N,n}\frac{Y_N^2-\mathbb{E }Y_N^2}{N^{n-\frac{1}{2}}} - \delta _{N,n}, \end{aligned}$$
(4.9)

where

  1. (i)

    \(\alpha _{N,n} \overset{a.s.}{\rightarrow } 1\) as \(N\rightarrow \infty \);

  2. (ii)

    \(\beta _{N,n} \overset{a.s.}{\rightarrow } \beta _n=\frac{2^{n-1}\Delta _{n,1}}{\Delta _{n,2}^2}\) as \(N\rightarrow \infty \);

  3. (iii)

    \(\delta _{N,n}\overset{a.s.}{\rightarrow } 0\) as \(N\rightarrow \infty \).

Moreover, for all \(p\geqslant 1\),

$$\begin{aligned} \sup _{N\geqslant n+4p-1} \max (\mathbb{E }|\alpha _{N,n}|^p, \mathbb{E }|\beta _{N,n}|^p, \mathbb{E }|\delta _{N,n}|^p) \leqslant C_{n,p}. \end{aligned}$$

The latter proposition is the first step in passing from the quotient \(Z_N=X_N/Y_N\) to the normalization required in Theorem 1.1. The fact that \(N^{n-\frac{1}{2}}\) appears in both of the denominators on the right-hand side of (4.9) indicates that both \(X_N\) and \(Y_N^2\) must be accounted for in order to capture the asymptotic normality of \(Z_N\).

Proof

Write

$$\begin{aligned} Z_N - \mathbb{E }Z_N&= \frac{X_N}{Y_N} - \frac{\mathbb{E }X_N}{\mathbb{E }Y_N} \\&= \frac{X_N - \mathbb{E }X_N}{Y_N} -\left( \frac{\mathbb{E }X_N}{\mathbb{E }Y_N}- \frac{\mathbb{E }X_N}{Y_N}\right) \\&= \frac{X_N - \mathbb{E }X_N}{Y_N} - \frac{(Y_N^2 - \mathbb{E }Y_N^2+\mathop {\mathrm{var}}(Y_N))\mathbb{E }X_N}{Y_N(Y_N+\mathbb{E }Y_N) \mathbb{E }Y_N } \\&= \frac{X_N - \mathbb{E }X_N}{Y_N} - \frac{(Y_N^2 - \mathbb{E }Y_N^2)\mathbb{E }X_N}{Y_N (Y_N+\mathbb{E }Y_N)\mathbb{E }Y_N } - \frac{\mathop {\mathrm{var}}(Y_N)\mathbb{E }X_N}{Y_N(Y_N+\mathbb{E }Y_N) \mathbb{E }Y_N }. \end{aligned}$$

Thus

$$\begin{aligned} \frac{Z_N-\mathbb{E }Z_N}{N^{\frac{n-1}{2}}} = \alpha _{N,n} \left( \frac{X_N-\mathbb{E }X_N}{N^{n-\frac{1}{2}}}\right) -\beta _{N,n} \left( \frac{Y_N^2-\mathbb{E }Y_N^2}{N^{n-\frac{1}{2}}}\right) -\delta _{N,n}, \end{aligned}$$

which shows that (4.9) holds with

$$\begin{aligned} \alpha _{N,n} = \frac{N^{\frac{n}{2}}}{Y_N}, \;\; \beta _{N,n} = \frac{N^{\frac{n}{2}}\mathbb{E }X_N}{Y_N(Y_N+\mathbb{E }Y_N)\mathbb{E }Y_N}, \;\; {\delta _{N,n}} = \beta _{N,n}\frac{\mathop {\mathrm{var}}(Y_N)}{N^{n-\frac{1}{2}}}. \end{aligned}$$

Using the factorization of \(Y_N\) in (4.6) and applying the SLLN for each \(\chi _k\) (\(k=N,\ldots ,N-n+1\)), we have

$$\begin{aligned} \frac{Y_N}{\sqrt{\frac{N!}{(N-n)!}}} \overset{a.s.}{\rightarrow } 1 \quad \text {as } N\rightarrow \infty , \end{aligned}$$

and hence

$$\begin{aligned} {\alpha _{N,n}}=\frac{N^{n/2}}{Y_N} \overset{a.s.}{\rightarrow } 1 \quad \text {as } N\rightarrow \infty . \end{aligned}$$

By the Cauchy-Binet formula (2.6) and the SLLN for U-statistics [(Theorem 3.1(2)], we have

$$\begin{aligned} \frac{1}{{N \atopwithdelims ()n}}Y_N^2 \overset{a.s.}{\rightarrow } \Delta _{n,2}^2 \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Thus

$$\begin{aligned} \beta _{N,n}= \frac{2^n{N\atopwithdelims ()n}\Delta _{n,1} }{Y_N^2 (1+\frac{\mathbb{E }Y_N}{Y_N})}\frac{N^{n/2}}{\mathbb{E }Y_N }\overset{a.s.}{\rightarrow } \frac{2^{n}\Delta _{n,1}}{2\Delta _{n,2}^2} \quad \text {as } N\rightarrow \infty . \end{aligned}$$

By Proposition 4.3(2), we also have \(\delta _{N,n}\overset{a.s.}{\rightarrow } 0 \text { as } N\rightarrow \infty \). To prove the last assertion, we note that for \(1\leqslant p\leqslant (N-n+1)/2\),

$$\begin{aligned} \mathbb{E }\left( \frac{N^{\frac{n}{2}}}{Y_N}\right) ^p \leqslant C_{n,p}, \end{aligned}$$

where \(C_{n,p}\) is a constant that depends on \(n\) and \(p\) only (see, e.g., [18, Lemma 4.2]).\(\square \)

Proof of Theorem 1.1

To simplify the notation, for \(I = \{i_1,\ldots ,i_n\}\subset \{1,\ldots ,N\}\), write \(d_I=|\mathop {\mathrm{det}}[g_{i_1}\cdots g_{i_n}]|\). Applying Proposition 4.4, we can write

$$\begin{aligned} \frac{Z_N-\mathbb{E }Z_N}{N^{\frac{n-1}{2}}} =\frac{{N\atopwithdelims ()n}}{N^{n-\frac{1}{2}}}(U_N -\mathbb{E }U_N) + A_{N,n} - B_{N,n} - \delta _{N,n}, \end{aligned}$$

where

$$\begin{aligned} U_N&= \frac{1}{{N\atopwithdelims ()n}} \sum _{|I|=n} (2^n d_I -\beta _n d_I^2),\\ A_{N,n}&= (\alpha _{N,n}-1)\left( \frac{X_N-\mathbb{E }X_N}{N^{n-\frac{1}{2}}}\right) , \end{aligned}$$

and

$$\begin{aligned} B_{N,n}=(\beta _{N,n}-\beta _n)\left( \frac{Y_N^2-\mathbb{E }Y_N^2}{N^{n-\frac{1}{2}}}\right) . \end{aligned}$$

Set \(I_0=\{1,\ldots ,n\}\). Applying Theorem 3.1(3) with

$$\begin{aligned} \zeta = \mathop {\mathrm{var}}(\mathbb{E }[ (2^n d_{I_0}-\beta _n d^2_{I_0})| g_1]), \end{aligned}$$
(4.10)

yields

$$\begin{aligned} \sqrt{N}\left( \frac{U_N - \mathbb{E }U_N}{n\sqrt{\zeta }}\right) \overset{d}{\rightarrow } \mathcal N (0,1) \quad \text {as } N\rightarrow \infty . \end{aligned}$$

By Proposition 4.4, \(\alpha _{N,n}\overset{a.s.}{\rightarrow } 1, \beta _{N,n}\overset{a.s.}{\rightarrow } \beta _n\) and \(\delta _{N,n}\overset{a.s.}{\rightarrow } 0\); moreover, each of the latter sequences is uniformly integrable. Thus by Hölder’s inequality and Proposition 4.2(1)

$$\begin{aligned} \mathbb{E }|A_{N,n}|\le (\mathbb{E }|\alpha _{N,n}-1|^2)^{1/2}C_{n}\rightarrow 0 \quad \text {as } N\rightarrow \infty . \end{aligned}$$

Similarly, using Proposition 4.3(1),

$$\begin{aligned} \mathbb{E }|B_{N,n}|\le (\mathbb{E }|\beta _{N,n}-\beta _n|^2)^{1/2}C_{n}\rightarrow 0 \quad \text {as } N\rightarrow \infty . \end{aligned}$$

By Slutsky’s theorem and the fact that \({N\atopwithdelims ()n}/N^n\rightarrow 1/n!\) as \(N\rightarrow \infty \), we have

$$\begin{aligned} \frac{n!(Z_N - \mathbb{E }Z_N)}{N^{\frac{n-1}{2}}n\sqrt{\zeta }} \overset{d}{\rightarrow } \mathcal N (0,1) \quad \text {as } N\rightarrow \infty . \end{aligned}$$
(4.11)

To conclude the proof of the theorem, it is sufficient to show that

$$\begin{aligned} \frac{n!\sqrt{\mathop {\mathrm{var}}(Z_N)}}{N^{\frac{n-1}{2}}n \sqrt{\zeta }} \rightarrow 1\quad \text {as } N\rightarrow \infty . \end{aligned}$$
(4.12)

Once again we appeal to uniform integrability: by Proposition 4.4,

$$\begin{aligned} \frac{|Z_N-\mathbb{E }Z_N|}{N^{\frac{n-1}{2}}} \leqslant 2^n|\alpha _{N,n}| \frac{|X_N-\mathbb{E }X_N|}{N^{n-\frac{1}{2}}} + |\beta _{N,n}|\frac{|Y_N^2 -\mathbb{E }Y_N^2|}{N^{n-\frac{1}{2}}}+ |\delta _{N,n}|. \end{aligned}$$

By Hölder’s inequality and Propositions 4.2(1), 4.3(1) and 4.4,

$$\begin{aligned} \sup _{N\geqslant n+8p-1} \left|\frac{Z_N-\mathbb{E }Z_N}{N^{\frac{n-1}{2}}}\right|^p \leqslant C_{n,p}, \end{aligned}$$

which, combined with (4.11), implies (4.12).\(\square \)