Some Remarks on Random Vectors and $O(n)$-Invariants

TL;DR

This paper explores the relationship between invariant random vectors and the theory of invariants, highlighting a specific connection involving the sum of basis elements and the expectation of a Veronese tensor.

Contribution

It presents new observations linking invariant random vectors to classical invariant theory, particularly relating sums of basis elements to expectations of Veronese tensors.

Findings

Sum of basis elements equals expectation of a Veronese tensor scaled by a known factor

Provides insights into the structure of $O(n)$-invariants

Connects invariant theory with probabilistic computations

Abstract

Computations involving invariant random vectors are directly related to the theory of invariants (cf. e.g \cite{Weing_1}). Some simple observations along these lines are presented in this paper. We note in particular that sum of elements of the standard basis of $ O(n)$-invariants is equal to the expectation of a random Veronese tensor up to a known scalar multiplier.

Equations

$$P^G(v)={\int }_{G}gvd\chi (g)$$

$$\mathbb{E}(x)=\mathbb{E}(P^G(x))$$

$$<x_1\otimes \cdots \otimes x_m,y_1\otimes \cdots \otimes y_m>=<x_1,y_1>\cdots <x_m,y_m>$$

$$v_1(J_1)\otimes \cdots \otimes v_k(J_k)$$

$$I(J_0)=\sum _{i=1}^n{e}_i(J_0)=\sum _{i=1}^n{e}_i\otimes e_i\in V\otimes V$$

$$I(\mathcal{P})=\sum _{i_1,i_2,\cdots ,i_k=1}^n{e}_{i_1}(J_1)\otimes e_{i_2}(J_2)\otimes \cdots e_{i_k}(J_k)$$

$$<I(\sigma {\mathcal{P}}_1),I({\mathcal{P}}_2)>=<I({\mathcal{P}}_1),I({\sigma }^{-1}{\mathcal{P}}_2)>$$

$$\sum _{\sigma \in {\mathfrak{S}}_k}\text{sign}(\sigma )I(\sigma {\mathcal{P}}_0)=0$$

$$S(V)\ni x\stackrel{\nu }{\to }{x}^{\otimes ,m}=x\otimes \cdots \otimes x\in S(V^{\otimes ,m})$$

$$\mathbb{E}(<x,y{>}^m)=\mathbb{E}(\parallel P^G(x^{\otimes ,m}){\parallel }^2)$$

$$\mathbb{E}(<x,y{>}^m)=\mathbb{E}(<x^{\otimes ,m},y^{\otimes ,m}>)=<\mathbb{E}(x^{\otimes ,m}),\mathbb{E}(y^{\otimes ,m})>=\mathbb{E}(\parallel P^G(x^{\otimes ,m}){\parallel }^2)$$

$$\mathbb{E}(x\otimes x)=\frac{1}{n}I(J_0)\equiv \frac{1}{n}\sum _{i=1}^n{e}_i\otimes e_i$$

$${\mathcal{U}}^{\times ,l}\ni (x_1,x_2,\cdots ,x_l)\to x_1(J_1)\otimes \cdots \otimes x_l(J_l)$$

$$\mathbb{E}(x(\mathcal{P}))=\frac{1}{n^l}I(\mathcal{P})$$

$${\mu }_{k,n}\equiv \mathbb{E}(<x,y{>}^{2k})=\frac{(2k-1)!!}{(n+2k-2)(n+2k-4)\cdots n}$$

$$A_m=\frac{1}{(m-1)!!}\sum _{\mathcal{P}\in {\Pi }_m}I({\mathcal{P}}_m)\in {\mathcal{I}}_m$$

$$\mathbb{E}(x^{\otimes ,m=2k})=\frac{1}{P(n,k)}\sum _{\mathcal{P}\in {\Pi }_m}I(\mathcal{P})=\frac{(2k-1)!!}{P(n,k)}{A}_m\equiv {\mu }_{k,n}{A}_m$$

$$P=\sum _{\mathcal{P}\in \Pi }{\alpha }_{\mathcal{P}}I({\mathcal{P}}_m)\in {\mathcal{I}}_m$$

$$<P-a^{\otimes ,m},I(\mathcal{P})>=0,\text{for all }\mathcal{P}\in {\Pi }_m$$

$$\mathcal{G}\alpha =1_L$$

$$L/N=\frac{1}{N^2}<\sum _{\mathcal{P}\in {\Pi }_m}I(\mathcal{P}),\sum _{\mathcal{P}\in {\Pi }_m}I(\mathcal{P})>=L/P(n,k)\equiv {\mu }_{k,n}$$

$$\mathbb{E}(X)=\left(\prod _{i=1}^l\frac{(2k_i-1)!!}{P(n,k_i)}\right)\frac{1}{|\Pi |}\sum _{({\mathcal{P}}_1,\cdots ,{\mathcal{P}}_l)\in \Pi }I({\mathcal{P}}_1)\otimes \cdots \otimes I({\mathcal{P}}_l)$$

$$\sum _{i=1}^{m_0}{s}_{i}I({\mathcal{P}}_i)={\mu }_{k,n}{A}_m$$

$$<I(\mathcal{P}),I(\mathcal{Q})>=n^l$$

$$\mathbb{E}(x(\mathcal{P}))=\frac{1}{n^k}I(\mathcal{P})\text{and}\mathbb{E}(y(\mathcal{Q}))=\frac{1}{n^k}I(\mathcal{Q})$$

$$\mathbb{E}(<x(\mathcal{P}),y(\mathcal{Q}>)=\prod _{i=1}^l{\int }_{S(V{)}^{|{\Theta }_i|}}\prod _{x_a,y_b\in {\Theta }_i}<x_a,y_b>d{\mu }_i$$

$$\frac{1}{n^{2k-l}}=\mathbb{E}(<x(\mathcal{P}),y(\mathcal{P})>)=\frac{1}{n^{2k}}<I(\mathcal{P}),I(\mathcal{Q})>$$

$$x=\{x_1,x_2,\cdots ,x_k\},y=\{y_1,y_2,\cdots ,y_k\}\in \mathcal{U}$$

$$\mathbb{E}(<x(\mathcal{P}),y(\mathcal{Q}>)=\frac{1}{n^{2k-|\mathcal{P}\vee \mathcal{Q}|}}$$

$$\mathcal{E}\equiv \mathbb{E}(x_{i_1,j_1}\cdots x_{i_m,j_m})=\mathbb{E}(<X{e}_{i_1},e_{j_1}>\cdots <X{e}_{i_m},e_{j_m})>$$

Full text

1 Introduction

We start with a brief overview of objects we will be dealing with (similar scenario was introduced in [5]). Let a compact Lie group $G$ act on a real vector space $V$. We will always assume that standard Euclidean scalar product $<,>$ on $V$ is $G$-invariant (cf. e.g [6]).

Let $\mathcal{U}\subset V$ be a $G$-invariant subset. A probability measure $\mu$ defined on $\mathcal{U}$ is called $G$-invariant if $\mu(g\mathcal{U}^{\prime})=\mu(\mathcal{U}^{\prime})$ for any measurable $\mathcal{U}^{\prime}\subset\mathcal{U}$ and any $g\in G$ (cf. e.g. [7], [8]). $\;\mathcal{U}$ will be called an invariant probability subspace of $V$ if such a measure exists.

A random vector $x$ distributed on $\mathcal{U}$ according to the invariant measure $\mu$ will be called $G$-invariant random vector or just invariant random vector.

The expectation of a random vector $x$ defined on a probability space $\mathcal{U}\subset V$ will be denoted by $\mathbb{E}_{\mu}(x)$ or just $\mathbb{E}(x)$. It is clear from definitions that for invariant random vectors $\mathbb{E}(gx)=\mathbb{E}(x)$.

Let $\chi$ be a normalized Haar measure on $G$ (cf. e.g. [6]-[8]). An invariant projector $P^{G}:V\rightarrow V^{G}$ onto the subspace of $G$-fixed points in $V$ can be thus written as

[TABLE]

for any $v\in V$ (cf. e.g. [6]).

The following well known statement is a starting point for this discussion (cf. e.g [1] or [5])

Lemma 1**.**

If $x$ is a random vector defined on any invariant probability subspace of a real representation $V$ of a compact (Lie) group $G$, then

[TABLE]

1.1 Tensors and Invariants

Let $G$ be a compact Lie group and $V$ be a (linear space of) real $G$-representation of dimension $n$. Let $V^{\otimes,m}$ be a space of $m$-tensors over $V$. Vector space $V^{\otimes,m}$ has a natural $G$-invariant dot-product defined by a rule :

[TABLE]

where $x_{1}\otimes\cdots\otimes x_{m},\;y_{1}\otimes\cdots\otimes y_{m}$ are any two decomposable tensors in $V^{\otimes,m}$. We fix once and for all an orthonormal basis $e_{1},e_{2},\cdots e_{n}$ in $V$.

Definition 1**.**

Let $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{k}\}$ be a partition of the set of indexes $J=\{1,2,\cdots,m\}$. For $v_{1},\cdots v_{k}\in V$ denote by

[TABLE]

a decomposable tensor in $V^{\otimes,m}$ that has vector $v_{1}$ at indexes enumerated by $J_{1}$, vector $v_{2}$ at positions enumerated by $J_{2}$ and so on.

For example, if $m=5,J_{1}=\{1,3,5\}$ and $J_{2}=\{2,4\}$ then $u(J_{1})\otimes v(J_{2})$ denotes the tensor $u\otimes v\otimes u\otimes v\otimes u$.

Remark 1**.**

The expression (1) should not be understood as a tensor product. It is used for notational purposes only. More standard way to specify tensors of this kind could employ Young tableau and permutation group, but the definition given above is sufficient for our purposes. In any case, assigning a meaning to an expression (1) is beyond the scope of this paper

We will always assume that all the parts of a partition $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{k}\}$ are non-trivial ($|J_{i}|\neq 0,\;j=1,\cdots k$) and we will use the notation $k=|\mathcal{P}|$ for the number of parts of the partition $\mathcal{P}$. If $m=2k$ and all the part sizes $|J_{i}|,\;i=1,\;\cdots,k$ are equal, the partition $\mathcal{P}$ is called pairing (cf. e.g. [1]- [4]). In case of $m=2$ there is just one pairing $J_{0}=(1,2),\;e_{i}(J_{0})=e_{i}\otimes e_{i},\;i=1,\cdots,n$ and there is a $G$-invariant tensor

[TABLE]

In general, for a pairing $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{k}\}$ on $J_{m=2k}$ there is a corresponding $G$-invariant tensor

[TABLE]

Let $\Pi_{m}$ denotes the set of all pairings on the set of indexes $J$. For the full orthogonal group $O(V)=O(n)$ the classical First Fundamental Theorem (FFT) of Invariant Theory (see [2] for example) actually states that

Theorem** **(FFT for Orthogonal Group).

There are no non-trivial $O(V)$ invariants in $V^{\otimes,m}$ if $m$ is odd and if $m=2k$, the invariants $I(\mathcal{P}),\mathcal{P}\in\Pi_{m}$ span the linear space $\mathcal{I}_{m}$ of $O(V)$ invariants in $V^{\otimes,m}$.

There are $(2k-1)!!\equiv 3\cdot 5\cdots(2k-1)$ pairings on the set of size $m=2k$. Take any pairing, e.g. $\mathcal{P}_{0}=(1,2)(3,4)\cdots(m-1=2k-1,m=2k)$. The permutation group $S_{m}$ of the index set $J_{m}$ naturally acts on the set of pairings $\Pi_{m}$ and it is clear that $\Pi_{m}$ is an $S_{m}$ orbit of $\mathcal{P}_{0}$. It should be also clear that

Lemma 2**.**

Given two pairings $\mathcal{P}_{1},\mathcal{P}_{2}\in\Pi_{m}$

[TABLE]

for any $\sigma\in S_{m}$.

Let $S_{k}\approx\mathfrak{S}_{k}\subset S_{m=2k}$ be a (sub)group of permutations of even-numbered indexes. It is easy to see, that if $k>n$ then

[TABLE]

where $\text{sign}(\sigma)=\pm 1$ is a sign of a permutation. All linear relations between $O(V)$ invariants are described by the classical Second Fundamental Theorem (SFT) (see for example a concise exposition in [2])

Theorem** **(SFT for Orthogonal Group).

The elements of the set of invariants $\{I(\mathcal{P}),\;\mathcal{P}\in\Pi_{m=2k}\}$ are linearly independent if $k\leq n$. For $k>n$ all linear relations between elements of this set are linear combinations of the identities of the form (2) above.

Definition 2**.**

Call the generating set of $O(n)$-invariants $I(\mathcal{P}),\mathcal{P}\in\Pi_{m=2k}$ the s̱tandard set (basis if $n\geq k$) of the space $\mathcal{I}_{m}$ of $O(V)$ invariants. The elements of this set will be called standard invariants

1.2 Random Tensors and Veronese Surface

For a unit sphere $S(V)\subset V$ one has an equivariant Veronese map

[TABLE]

The image $\mathfrak{V}_{m}$ of the map $\nu$ is called Veronese surface (cf. e.g. [11]). If $\mathcal{U}\subset S(V)$ is a $G$-invariant probability subspace in $S(V)$ then $\nu(\mathcal{U})\subset\mathfrak{V}_{m}$ is an invariant probability subspace of the Veronese surface (cf. e.g. [5]) and for an invariant random unit vector $x\in\mathcal{U}$ there is a corresponding invariant random Veronese tensor $x^{\otimes,m}\in\mathcal{U}^{\otimes,m}\subset\mathfrak{V}_{m}$. As in [5] we have

Lemma 3**.**

For independent invariant random vectors $x,y$ defined on any invariant probability subspace $\mathcal{U}\subset S(V)$

[TABLE]

Proof. By Lemma 1 and independence of $x$ and $y$

[TABLE]

From Lemma 1 (see [5] for details), we have also

Lemma 4**.**

If the representation of $G$ in $V$ is irreducible then for a random vector $x$ defined on any invariant probability subspace $\mathcal{U}\subset S(V)$

[TABLE]

Moreover, in accordance with Definition 1 (cf. (1)), for any partition $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{l}\}$ of the set of indexes $J=\{1,2,\cdots,m\}$ and for any invariant probability subspace $\mathcal{U}\subset S(V)$ there is a generalized Veronese surface defined as an image of the map

[TABLE]

and applying Lemmas 1 and 4 we get

Lemma 5**.**

For any set of independent random vectors $(x_{1},x_{2},\cdots x_{l})\in\mathcal{U}$ defined on any invariant probability subspace $\mathcal{U}\subset S(V)$ and for any pairing $\mathcal{P}\in\Pi_{2l}$, let $x(\mathcal{P})$ be a random Veronese tensor defined by a right hand side of (4). If representation of $G$ in $V$ is irreducible then

[TABLE]

2 Expectations and Invariants

From now on, unless explicitly stated otherwise, we assume that the group $G$ is a full orthogonal group $O(V)\approx O(n)$. For obvious reasons, everything that was said above remains valid in this special case. Note that $S(V)$ is an orbit of $O(V)$ and hence the $O(V)$-invariant probability measure on a unit sphere $S(V)$ is uniquely inherited from the Haar measure on $O(n)$ (cf. e.g. [8]). By direct computation (see for example [9], [10]) one gets

Lemma 6**.**

For independent invariant random vectors $x,y$ defined on $S(V)$

[TABLE]

Let $m=2k$. Recall, that the space $\mathcal{I}_{m=2k}$ is spanned by the set of standard invariants $\mathcal{V}=\{I(\mathcal{P}),\mathcal{P}\in\Pi_{m}\}$. Let

[TABLE]

be the average of the set of standard invariants in $\mathcal{I}_{m}$. Denote the denominator on the right hand side of (5) by $P(n,k)$ and let $\mathcal{G}$ denote the Gram matrix of the ordered standard set of invariants $\mathcal{V}$.

Theorem 1**.**

($m=2k$)

1. (i)

The orthogonal projection of the Veronese surface $\mathfrak{V}_{m}$ onto $\mathcal{I}_{m}$ is just one point $\mu_{k,n}A_{m}$. 2. (ii)

Sum of the elements of any row of $\mathcal{G}$ is equal to $P(n,k)$ and the average of elements of $\mathcal{G}$ is equal to $\frac{1}{\mu_{k,n}}$ 3. (iii)

Therefore, the expectation of a random Veronese tensor $x^{\otimes,2k}$ is an average of elements of the standard basis of $O(V)$-invariants divided by the average of the matrix elements of $\mathcal{G}$, in other words

[TABLE]

Proof. Since $\mathfrak{V}_{m}$ is an orbit of $S(V)$, its (equivariant) orthogonal projection onto $\mathcal{I}_{m}$ consists of just one point. To find this point, let

[TABLE]

be a projection of Veronese tensor $a^{\otimes,m},\;a\in S(V)$ onto $\mathcal{I}_{m}$ where $\alpha_{\mathcal{P}},\mathcal{P}\in\Pi_{m}$ are real numbers that must satisfy the ”normal” equations

[TABLE]

It is obvious that $<a^{\otimes,m},I(\mathcal{P})>\;=\;1$ for all $\mathcal{P}\in\Pi_{m}$. Using an appropriate order on the set of pairings rewrite this system of linear equations as

[TABLE]

where $L=(2k-1)!!,\;\alpha$ is an unknown vector with coordinates $\alpha_{\mathcal{P}},\;\mathcal{P}\in\Pi_{m}$ and $1_{L}$ is the $L$-dimensional vector of ones. Since $\mathcal{V}$ is an $S_{m}$-orbit, all row-sums of $\mathcal{G}$ are equal to each other by Lemma 2. Denoting the unique row-sum of $\mathcal{G}$ by $N$ we see that vector $(1/N)1_{L}$ is a solution of the system of equations (6). The value of $N$ can be easily found by comparing (3) and (5). Indeed, from Lemma 3 (with $m=2k$) and (5), (6) one gets

[TABLE]

It follows that $N=P(n,k)$ and that the number $1/\mu_{k,n}$ is the average of elements of the Gram matrix $\mathcal{G}$. That proves (i) and (ii) while (iii) is equivalent to (i) by Lemma 1.

It is now a straightforward exercise to verify the following corollary

Theorem 2**.**

Let $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{l}\}$ be a partition of the set of indexes $J=\{1,2,\cdots,m\}$, let $x_{1},x_{2},\cdots x_{l}\in S(V)$ be independent $O(V)$-invariant random vectors and let $X=x_{1}(J_{1})\otimes\cdots\otimes x_{l}(J_{l})$ be a generalized random Veronese tensor as in (4)

1. (i)

If cardinality of at least one of the sets $J_{i},\;i=1,\cdots l$ is odd then $\mathbb{E}(X)=0$ 2. (ii)

If $|J_{i}|=2k_{i},\;i=1,\cdots l$, let $\Pi(J_{i})$ be the set of all $(2k_{i}-1)!!$ pairings on the set $J_{i},\;i=1,\cdots l$ and let $\Pi=\Pi(J_{1})\times\cdots\times\Pi(J_{l})$. Then in notation of Definition 1 (cf. Remark 1)

[TABLE]

Corollary 1**.**

Let $\mathcal{V}_{0}=\{I(\mathcal{P}_{i}),\mathcal{P}_{i}\in\Pi^{\prime}\subset\Pi_{m}\},\;i=1,\cdots,m_{0}\leq m$ be an ordered basis of $\mathcal{I}_{m=2k}$ and let $\mathcal{G}_{0}$ be the corresponding Gram matrix of $\mathcal{V}_{0}$. Let $s_{i},\;i=1,\cdots,m_{0}$ be the sum of elements of the $i$-th row of $\mathcal{G}^{-1}_{0}$. Then the following identity holds

[TABLE]

Proof. Following the proof of Theorem bear in mind that: (a) projection of the Veronese surface onto $\mathcal{I}_{m}$ belongs to the linear span of $\mathcal{V}_{0}$, and (b), that $(s_{1},\cdots,s_{m_{0}})^{T}=\mathcal{G}^{-1}_{0}1_{m_{0}}$

Remark 2**.**

Finding explicit values of $s_{i}$ in case of $n<k$ is probably a non-trivial problem. A method for selecting a basis from the standard set of invariants can be found in [3]

2.1 Gram Matrix of the Standard Set of Invariants

For any two partitions $\mathcal{P}_{1},\mathcal{P}_{2}$ denote their least upper bound by $\mathcal{P}_{1}\lor\mathcal{P}_{2}$. The following useful statement is well known (cf. e.g. [1] and [4]).

Lemma 7**.**

Let $\mathcal{P}=\{P_{1},P_{2},\cdots,P_{k}\}$ and $\mathcal{Q}=\{Q_{1},Q_{2},\cdots,Q_{k}\}$ be two pairings of the index set $J=\{1,2,\cdots,m=2k\}$, let $\mathcal{R}=\{R_{1},R_{2},\cdots,R_{l}\}=\mathcal{P}\lor\mathcal{Q}$ and set $l=|\mathcal{P}\lor\mathcal{Q}|$. Then

[TABLE]

It is not hard to come up with a combinatorial proof of this statement. We prefer, however, to verify this fact by computing expectations. By Lemma 5, there are generalized random Veronese tensors $x(\mathcal{P})=x_{1}(P_{1})\otimes\cdots\otimes x_{k}(P_{k})$ and $y(\mathcal{Q})=y_{1}(Q_{1})\otimes\cdots\otimes y_{k}(Q_{k})$ such that

[TABLE]

One can further assume that random unit vectors $x_{i},y_{i},i=1,\cdots,k$ are pairwise independent. By separating variables we have

[TABLE]

where sets of variables $\Theta_{i}$ are pairwise disjoint, $|\Theta_{i}|=|R_{i}|,i=1,\cdots,l$ and every variable occurs exactly twice in each of the integrals over a product of $|\Theta_{i}|$ unit spheres (denoted by $S(V)^{|\Theta_{i}|}$). In vector coordinates, the $i$-th integral splits into a sum of integrals of monomials. Among these monomials there are exactly $n$ that are products of $|R_{i}|$ squares and the rest of the monomials will have at least one independent multiple of the form $x_{i}x_{j},i\neq j$ (two different coordinates of the same vector). It is easy to see (cf. Lemma 4 and [5]) that square terms will contribute a value of $n^{-|R_{i}|}$ each, while all other terms will vanish. Hence, the right hand side of (9) evaluates to $n^{-(2k-l)}$ and then it follows from (8) that

[TABLE]

which is the same as (7).

By Lemma 7 every entry of the Gram matrix $\mathcal{G}$ is a power of $n$ and by Theorem 1 (ii), the sum of entries of every row of $\mathcal{G}$ is equal to $P(n,k)=n^{k}+c_{1}n^{k-1}+\cdots c_{k}n$ which is a value of the polynomial of $n$ with fixed integer coefficients (that do not depend on $n$). Therefore, every row of $\mathcal{G}$ has exactly one element equal to $n^{k}$, $c_{1}$ elements equal to $n^{k-1}$ and so on. It is easy to see, therefore, that the following statement is true

Corollary 2**.**

For $t=1,\;3,\;5,\;\cdots,\;2k-1$ let $M_{t}$ be a $t\times t$ matrix with all its diagonal elements equal to $n$ and all its off-diagonal elements equal to one. Let $M$ be the Kronecker product of matrices $M_{t},\;t=1,\;3,\;\cdots,\;2k-1$.Then, the rows of $\mathcal{G}$ and $M$ are the same up-to a (row dependent) permutation.

Another straightforward application of Lemmas 1, 5 and 7 yields

Corollary 3**.**

Let $G$ be a compact Lie group and let $V$ be a real $G$-representation. For any set of $2k$ independent random $G$-invariant vectors

[TABLE]

defined on any $G$-invariant probability subspace $\mathcal{U}\subset S(V)$ and for any pairings $\mathcal{P},\mathcal{Q}\in\Pi_{2k}$, let $x(\mathcal{P}),y(\mathcal{P})$ be corresponding generalized random Veronese tensors in $\mathcal{U}^{\otimes,m}$ defined as in (4). If representation of $G$ in $V$ is irreducible then

[TABLE]

2.2 Weingarten Calculus

Let $X$ be a random Haar-distributed matrix in $O(V)\approx O(n)$. One of the standard problems addressed by Weingarten calculus (see e.g. [1] and references therein) is computation of moments $\mathbb{E}(x_{i_{1},j_{1}}\cdots x_{i_{m},j_{m}})$ where $x_{i,j}$ is an $(i,j)$-entry of $X$. To apply Theorem 2 in this context, recall that $x_{i,j}=\;<Xe_{i},e_{j}>$ and hence (cf. e.g. [1])

[TABLE]

Let’s $\{i^{\prime}_{1},\cdots i^{\prime}_{l}\}$ be the set of all distinct first indices in (10). Suppose that $i^{\prime}_{t}$ occurs in the sequence $i_{1}\cdots i_{m},\;r_{t}$ times, $\sum_{t=1}^{l}r_{t}=m$. Let $\mathcal{P}=\{J_{1},J_{2},\cdots,J_{l}\}$ be a partition of the set of indexes $J=\{1,2,\cdots,m\}$ such that $J_{1}$ consists of the first $r_{1}$ elements of $J,\;J_{2}$ consists of the next $r_{2}$ elements of $J$ and so on. Rearranging first indexes on the right hand side of (10) in accordance with the partition $\mathcal{P}$ gives rise to a permutation of the corresponding sequence of second indexes in dot-product terms. Let thus permuted second index sequence be $\sigma=(j^{\prime}_{1},\cdots j^{\prime}_{m})$. Split the sequence $\sigma$ into subsequences of $\sigma_{1},\sigma_{2},\cdots,\sigma_{l}$ so that $\sigma_{1}$ contains first $r_{1}$ elements of $\sigma$, $\sigma_{2}$ contains next $r_{2}$ elements of $\sigma$ and so on. Further, for $\sigma_{i}=(a_{1},a_{2},\cdots,a_{r_{i}})$ set $e(\sigma_{i})=e_{a_{1}}\otimes e_{a_{2}}\otimes\cdots\otimes e_{a_{r_{i}}},\;i=1,\cdots,l$. The next step is to form a random $O(n)$-invariant Veronese tensor (cf. (4))

[TABLE]

The columns $Xe_{i^{\prime}_{t}},\;t=1,\cdots l$ of the matrix $X$ are not independent as random vectors. However, these columns are pairwise orthogonal and hence $<I(\mathcal{Q}),\mathcal{X}>\;=\;0$ for any pairing $\mathcal{Q}$ on $J$ that is not a refinement of the partition $\mathcal{P}$. Therefore, applying the equivariant projector $P^{O(V)}:V^{\otimes,m}\rightarrow\mathcal{I}_{m}$ from Lemma 1 to the generalized random Veronese tensor (11) and arguing as in the proof of Theorem 1, we get

[TABLE]

where $P_{t}^{O(V)}$ is an equivariant projector $P_{t}^{O(V)}:V^{\otimes,k_{t}}\rightarrow\mathcal{I}_{k_{t}}$, and by Lemma 1 the right hand side of (12) is equal to

[TABLE]

Assuming introduced notation we have a computational recipe.

Theorem 3**.**

(Cf. Theorem 2). Expectation $\mathcal{E}$ in (10) can be computed as follows:

1. (i)

If $|J_{i}|$ is odd for at least one of the sets $J_{i},\;i=1,\cdots l$ then $\mathcal{E}=0$ 2. (ii)

If $r_{i}=|J_{i}|=2k_{i},\;i=1,\cdots l$, let $A_{i}$ be an average of standard $2k_{i}\;\;O(V)$-invariants defined by $(2k_{i}-1)!!$ pairings of the set $J_{i},\;i=1,\cdots l$. Then

[TABLE]

Corollary 4**.**

If indexes $i_{1},\cdots i_{l}$ are pairwise distinct then

[TABLE]