2 Laws of large numbers

In this section, we state the three main Laws of Large Numbers. Our approach mostly follows [Reference Billingsley1, Reference Durrett5, Reference Murty21].

First of all, in heuristic terms, the Weak Law of Large Numbers says that if n is large, then there is only a small chance that the fraction of heads in n tosses will be far from $1/2$ . In more precise terms, it is formulated in the following way.

Theorem 2.1 (Weak Law of Large Numbers; cf. [Reference Billingsley1, p. 86], [Reference Durrett5, Theorem 2.2.12])

Let $X_1, X_2,\dots $ be a sequence of independent and identically distributed simple random variables with identical expected values

$$ \begin{align*}E(X_n) = \mu\text{,}\end{align*} $$

for each n. Put

$$ \begin{align*}S_n = X_1 + \cdots + X_n\text{.}\end{align*} $$

Suppose that $y> 0$ . Then

$$ \begin{align*}\lim_{n \to \infty} \operatorname{Prob}\left( \left| n^{-1} S_n - \mu \right| \geqslant y \right) = 0 \text{.} \end{align*} $$

Turning to the Strong Law of Large Numbers, recall that it expands upon the Weak Law (Theorem 2.1).

Theorem 2.2 (Strong Law of Large Numbers; cf. [Reference Billingsley1, Theorem 6.1], [Reference Durrett5, Theorem 2.4.1])

Let $X_1, X_2,\dots $ be a sequence of independent and identically distributed simple random variables with identical expected values

$$ \begin{align*}E(X_n) = \mu\text{,}\end{align*} $$

for each n. Put

$$ \begin{align*}S_n = X_1 + \ldots + X_n\text{.}\end{align*} $$

It then holds true that

$$ \begin{align*}\operatorname{Prob}\left(\lim_{n \to \infty} n^{-1} S_n = \mu \right) = 1 \text{.} \end{align*} $$

Finally, we state the Central Limit Theorem. It also extends the Weak Law of Large Numbers. An attractive self-contained proof via the theory of Fourier transforms is given in [Reference Murty21, Section 3.15].

Theorem 2.3 (Central Limit Theorem; cf. [Reference Billingsley1, Theorem 27.1], [Reference Durrett5, Theorem 3.4.1], [Reference Murty21, Theorem 3.16])

Let $\Phi $ be a standard normal random variable and so, in particular, having probability density function

$$ \begin{align*}\phi(y) = \frac{1}{\sqrt{2 \pi}} e^{- \frac{y^2}{2} } \text{.} \end{align*} $$

Let $X_1, X_2,\dots $ be a sequence of independent random variables having the same distribution with mean $\mu $ and positive finite variance $\sigma ^2$ . Let

$$ \begin{align*}S_n = X_1 + \cdots + X_n\text{.}\end{align*} $$

It then holds true that

$$ \begin{align*}\lim_{n \to \infty} \frac{S_n - n \mu}{\sigma \sqrt{n}} \to \Phi \text{.} \end{align*} $$

Furthermore,

$$ \begin{align*}\lim_{n \to \infty} n^{-1} S_n \to \mu \text{.} \end{align*} $$

3 The theory of Harder and Narasimhan

Here, we recall, from [Reference Harder and Narasimhan18], the theory of Harder and Narasimhan. In what follows, C denotes a nonsingular projective algebraic curve over an arbitrary characteristic algebraically closed base field. By abuse of terminology, we fail to distinguish among the concept of finite rank locally free ${\mathcal O}_C$ -modules and total spaces of vector bundles. On the other hand, if $\mathcal {E}$ is a vector bundle on C and $\mathcal {F}$ a locally free submodule, then $\mathcal {F}$ is called a subvector bundle if the quotient $\mathcal {E} / \mathcal {F}$ is locally free.

If $\mathcal {E}$ is a nonzero vector bundle on C, then its slope is

$$ \begin{align*}\mu(\mathcal{E}) := \frac{\operatorname{deg}\left(\mathcal{E} \right)}{\operatorname{rank}\left(\mathcal{E}\right)} = \frac{ \operatorname{deg}\left( \operatorname{det}\left(\mathcal{E}\right)\right)}{r} \text{.} \end{align*} $$

Here,

$$ \begin{align*}r := \operatorname{rank}\left(\mathcal{E}\right)\end{align*} $$

is the rank of $\mathcal {E}$ and

$$ \begin{align*}\operatorname{det}\left(\mathcal{E}\right) := \bigwedge^r\left(\mathcal{E}\right) \end{align*} $$

is the determinant line bundle.

If a nonzero vector bundle $\mathcal {E}$ on C has the property that

$$ \begin{align*}\mu(\mathcal{F}) \leqslant \mu(\mathcal{E}) \text{,} \end{align*} $$

for all nonzero proper subbundles $\mathcal {F}$ , then it is called semistable. By [Reference Harder and Narasimhan18, Lemma 1.3.7], if a nonzero vector bundle $\mathcal {E}$ is not semistable, then it admits a unique nonzero subbundle $\mathcal {F}$ which strongly contradicts semistability in the following sense:

1. (i) $\mathcal {F}$ is semi-stable; and

2. (ii) if $\mathcal {G}$ is a subbundle of $\mathcal {E}$ which contains $\mathcal {F}$ as a subbundle, then

   $$ \begin{align*}\mu(\mathcal{F})> \mu(\mathcal{G})\text{.}\end{align*} $$

The concept of subbundles which strongly contradict semistability is a key technical point in establishing the existence and uniqueness of the Harder and Narasimhan (canonical) filtrations. This fundamental result of [Reference Harder and Narasimhan18] is formulated in the following way.

Theorem 3.1 [Reference Harder and Narasimhan18, Section 1.3]

Let C denote a nonsingular projective algebraic curve over an arbitrary characteristic algebraically closed base field. Let $\mathcal {E}$ be a nonzero vector bundle on C. Then $\mathcal {E}$ admits a uniquely determined flag of subvector bundles

(3.1) $$ \begin{align} 0 = \mathcal{E}_0 \subsetneq \mathcal{E}_1 \subsetneq \cdots \subsetneq \mathcal{E}_\ell = \mathcal{E}, \end{align} $$

which has the two properties that:

1. (i) the successive quotients $\mathcal {E}_i / \mathcal {E}_{i-1}$ are semistable for $i = 1,\dots ,\ell $ ; and

2. (ii) the slopes of the successive quotients are strictly decreasing in the sense that

   $$ \begin{align*}\mu(\mathcal{E}_i / \mathcal{E}_{i-1})> \mu(\mathcal{E}_{i+1} / \mathcal{E}_i), \end{align*} $$
   for $i = 1,\dots ,\ell -1$ .

In working with the Harder and Narasimhan filtrations (3.1), in what follows, we find it useful to put

$$ \begin{align*}\mu_i := \mu(\mathcal{E}_i / \mathcal{E}_{i-1}) \text{ and } r_i := \operatorname{rank}(\mathcal{E}_i / \mathcal{E}_{i-1}) ,\end{align*} $$

for $i = 1,\dots ,\ell $ . By this notation $r_i> 0\text {,}$ for all $i = 1,\dots ,\ell $ ,

$$ \begin{align*}\mu_1> \cdots > \mu_\ell \end{align*} $$

and

$$ \begin{align*}\sum_{i=1}^\ell r_i = r \text{.} \end{align*} $$

Moreover, if

$$ \begin{align*}\mu := \mu\left(\mathcal{E}\right) \text{, } \end{align*} $$

then

$$ \begin{align*}\mu = \frac{ \sum_{i=1}^\ell \mu_i r_i }{r} \text{.} \end{align*} $$

Finally, we refer to the bundles $\mathcal {E}_i$ , for $i = 1,\dots ,\ell $ , which arise in the filtration (3.1), as the Harder and Narasimhan subbundles of $\mathcal {E}$ .

4 The Harder and Narasimhan polygons and their vertices

A concept of Harder and Narasimhan Polygons emerged as a tool for expanding upon Harder and Narasimhan’s theory of filtrations for vector bundles on curves. This was popularized by Grayson in his work on lattice reduction theory [Reference Grayson10]. We refer to the article [Reference Casselman2], by Casselman, for an exposition. Here, our viewpoint is to associate a polygon with Harder and Narasimhan data. We make this precise in Definition 4.1.

Another context in which a fruitful theory of Narasimhan Polygons has emerged is that of filtered vector spaces. Such developments have been made possible by work of Faltings and Wüstholz [Reference Faltings and Wüstholz8], Chen [Reference Chen3], and others. In Example 4.2, and because of its relevance to Theorem 1.5, we indicate how our notion of Harder and Narasimhan data fits within the framework of the Harder and Narasimhan filtration that is associated with each filtered vector space.

Within the context of Diophantine approximation, the theory of Harder and Narasimhan filtrations for vector spaces is an important aspect of the work of Faltings and Wüstholz [Reference Faltings and Wüstholz8]. Similar filtrations together with a theory of polygons arise, more recently, in the work of Evertse and Ferretti (see [Reference Evertse and Ferretti6, Section 15]).

Of interest, for our purposes here, is the asymptotic probabilistic nature of such Harder and Narasimhan Polygons. Exactly what is meant by this is made precise below.

Definition 4.1 By Harder and Narasimhan data, is meant a collection of strictly decreasing rational numbers

$$ \begin{align*}\mu_1> \cdots > \mu_\ell\end{align*} $$

together with a collection of positive integers $r_i> 0$ , for $i = 1,\dots ,\ell $ . Denote such data by $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ . Here,

$$ \begin{align*}\overrightarrow{\mu} := (\mu_1,\dots,\mu_\ell) \text{ and } \overrightarrow{r} := (r_1,\dots,r_\ell)\text{.}\end{align*} $$

The integer $\ell $ is called the length.

Put

$$ \begin{align*}r := \sum_{i=1}^\ell r_i\text{.}\end{align*} $$

This is the rank, and the $r_i$ are the subquotient ranks.

Finally, upon setting

$$ \begin{align*}d_i := r_i \mu_i\text{,}\end{align*} $$

for $i = 1,\dots ,\ell $ , and

$$ \begin{align*}\overrightarrow{d} := (d_1,\dots,d_\ell)\text{,}\end{align*} $$

we obtain the degree vector. If

$$ \begin{align*}\left| \overrightarrow{d} \right| := \sum_{i=1}^\ell d_i> 0 \text{,}\end{align*} $$

then the Harder and Narasimhan data $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ are said to have positive degree.

Furthermore, set

$$ \begin{align*}\overrightarrow{p} := (p_1,\dots,p_\ell), \end{align*} $$

where

$$ \begin{align*}p_i := \frac{r_i}{r} \text{,} \end{align*} $$

for $i = 1,\dots ,\ell $ . Then the $p_i$ are the Harder and Narasimhan probabilities, whereas $\overrightarrow {p}$ is the Harder and Narasimhan probability vector.

Finally, similar to the approach of [Reference Grayson10, Definition 1.10 and Discussion 1.16] and [Reference Casselman2, p. 630], the vertices of Harder and Narasimhan data $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r} \right )$ are defined to be the collection of points

$$ \begin{align*}\operatorname{Verticies}\left(\overrightarrow{r}, \overrightarrow{d}\right) := \left\{ (r_i, d_i) : i = 1,\dots,\ell \right\} \text{.} \end{align*} $$

(Compare also with [Reference Chen3, Remark 2.2.8].)

Example 4.2 As in [Reference Faltings and Wüstholz8, Section 4], Let V be a finite-dimensional $\mathbf {k}$ -vector space, and consider, given a collection of real numbers

$$ \begin{align*}0 \leqslant \lambda_0 < \lambda_1 < \cdots < \lambda_n < \lambda_{n+1}\text{,} \end{align*} $$

a filtration of the form

(4.1) $$ \begin{align} V = F^{\lambda_0} V \supsetneq F^{\lambda_1} V \supsetneq \cdots \supsetneq F^{\lambda_n} V \supsetneq F^{\lambda_{n +1}}V = 0 \text{.} \end{align} $$

The filtered vector space (4.1) is called semistable if it holds true that

(4.2) $$ \begin{align} \mu(V') \leqslant \mu(V) \end{align} $$

for each proper nonzero subspace

$$ \begin{align*}0 \not = V' \subsetneq V \text{.} \end{align*} $$

In (4.2), the slope of $V'$ is calculated with respect to the induced filtration.

As noted by Faltings and Wüstholz, each such filtered vector space (4.1) admits a canonical Harder and Narasimhan filtration. In more specific terms, there exists a flag of vector spaces

$$ \begin{align*}0 = V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_{\ell} = V \text{,} \end{align*} $$

which have the property that each successive quotient $V_i / V_{i - 1}$ , for $i = 1,\dots ,\ell $ , is semistable (which respect to the induced filtration), and furthermore, the slopes $\mu (V_i / V_{i-1})$ of the successive quotients are strictly decreasing.

Within this context, putting

$$ \begin{align*}\mu_i := \mu(V_i / V_{i-1}) \text{ and } r_i := \dim(V_i / V_{i-1}) \text{,} \end{align*} $$

for $i = 1,\dots ,\ell $ , and setting

$$ \begin{align*}\overrightarrow{\mu} := (\mu_1,\dots,\mu_\ell) \text{ and } \overrightarrow{r} := (r_1,\dots,r_\ell) \end{align*} $$

yields Harder and Narasimhan data $\operatorname {HN}(\overrightarrow {\mu },\overrightarrow {r})$ .

5 Tensor products of Harder and Narasimhan subbundles

To provide motivation for our construction with vertices of Harder and Narasimhan subbundles in Section 6, here we discuss a related construction of [Reference Codogni and Patakfalvi4], which was the starting point for our investigation here.

These constructions from [Reference Codogni and Patakfalvi4] build on Viehweg’s fiber product approach from [Reference Viehweg25], for proving weak positivity results for direct images of tensor powers of relative canonical sheaves.

A representative example for the techniques of [Reference Codogni and Patakfalvi4, Section 5] is explained in [Reference Codogni and Patakfalvi4, Remark 5.6]. We reproduce some of that discussion here. It provides motivation for our analogous results which apply to the context of filtered vector spaces. (See Theorem 1.5 and Example 4.2.)

In particular, working over the projective line $\mathbb {P}^1_{\mathbf {k}}$ , consider the case of a vector bundle

$$ \begin{align*}\mathcal{E} := \bigoplus_{1 \leqslant i \leqslant \ell} {\mathcal O}_{\mathbb{P}^1_{\mathbf{k}}}(b_i)^{\oplus n_i}, \end{align*} $$

where $b_i \in \mathbb {Z}$ and $b_i < b_{i+1}\text {,}$ for $i = 1,\dots ,\ell -1$ .

With respect to the Harder and Narasimhan filtrations for $\mathcal {E}$ , the ith Harder and Narasimhan submodule is

$$ \begin{align*}\mathcal{E}_i := \bigoplus_{1 \leqslant j \leqslant i} {\mathcal O}_{\mathbb{P}^1}(b_j)^{\oplus n_j} \text{.} \end{align*} $$

The ith subquotient slope is thus

$$ \begin{align*}\mu_i := \mu\left( \mathcal{E}_i / \mathcal{E}_{i-1} \right) = b_i \text{.} \end{align*} $$

Now, fixing a positive integer $m> 0$ , the idea is to study via probabilistic methods, inside of $\mathcal {E}^{\otimes m}$ , the prevalence of those submodules which are obtained via the m-fold tensor products

$$ \begin{align*}\bigotimes_{i=1}^m \mathcal{E}_{a_i} \subseteq \mathcal{E}^{\otimes m} \text{.} \end{align*} $$

Here, $a_i \in \{1,\dots ,\ell \}\text {,}$ for $i = 1,\dots ,m$ , and

$$ \begin{align*}\sum_{i=1}^m \mu_{a_i} \geqslant z \text{,} \end{align*} $$

for $z \in \mathbb {Z}_{\geqslant 0}$ some given nonnegative integer.

In Section 6, we generalize this construction from [Reference Codogni and Patakfalvi4], so as to treat the more general context of Harder and Narasimhan vertices.

6 Statement of main theorem and its proof

In order to state our main theorem (see Theorem 6.1), let us first consider the construction from Section 5 in a slightly more general context.

Let $[\ell ]:= \{1,\dots ,\ell \}\text {.}$ Fixing a positive integer $m \in \mathbb {Z}_{>0}$ , let $[\ell ]^m$ be the m-fold Cartesian product of $[\ell ]$ .

Let $\succ $ be the partial order on $[\ell ]^m$ that is defined in the following way. If $\mathbf {a}, \mathbf {b} \in [\ell ]^m\text {,}$ then $\mathbf {a} \succeq \mathbf {b}$ if $a_j \geqslant b_j$ for all $j = 1,\dots ,m$ , whereas $\mathbf {a} \succ \mathbf {b}$ if $\mathbf {a} \succeq \mathbf {b}$ and $a_j> b_j$ for some $j = 1,\dots , m$ .

Now, fix Harder and Narasimhan data $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ . Of particular interest is the case that the Harder and Narasimhan data $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ have positive degree. However, on the other hand, we do not impose that condition for the present time being.

Inside of $[\ell ]^m$ , define the subset $S_{m,z}(\overrightarrow {\mu })$ , for $z \in \mathbb {Z}_{\geqslant 0}$ a fixed nonnegative integer, by the condition that

$$ \begin{align*}S_{m,z}\left(\overrightarrow{\mu}\right) := \left\{ \mathbf{a} = (a_1,\dots,a_m) \in [\ell]^m : \sum_{j=1}^m \mu_{a_j} \geqslant z \right\} \text{.} \end{align*} $$

If $ \mathbf {a} \in [\ell ]^m \text {,} $ then put

$$ \begin{align*}v_{\overrightarrow{\mu}}(\mathbf{a}) := \sum_{j=1}^m \mu_{a_j} \text{.} \end{align*} $$

The set $S_{m,z}\left (\overrightarrow {\mu }\right )$ has the following two properties:

1. (i) if $\mathbf {a} \in S_{m,z}(\overrightarrow {\mu })\text {,}$ $\mathbf {b} \in [\ell ]^m$ , and $\mathbf {a} \succeq \mathbf {b}\text {,}$ then $\mathbf {b} \in S_{m,z}(\overrightarrow {\mu })\text {;}$ and

2. (ii) if $\mathbf {a} \in [\ell ]^m\text {,}$ then $v_{\overrightarrow {\mu }}(\mathbf {a}) \geqslant z\text {,}$ if and only if $\mathbf {a} \in S_{m,z}(\overrightarrow {\mu })\text {.}$

Henceforth, put

$$ \begin{align*}d := \# S_{m,z}(\overrightarrow{\mu}) \text{.} \end{align*} $$

Then, upon arranging the elements of $[\ell ]^m$ in decreasing order, with respect to $v_{\overrightarrow {\mu }}(\cdot )$ , we may write

$$ \begin{align*}[\ell]^m = S_{m,z}(\overrightarrow{\mu}) \bigsqcup \{\mathbf{a}_{d+1},\dots,\mathbf{a}_e \}, \end{align*} $$

where

$$ \begin{align*}S_{m,z}(\overrightarrow{\mu}) = \{\mathbf{a}_1,\dots,\mathbf{a}_d\} \text{,} \end{align*} $$

$$ \begin{align*}v_{\overrightarrow{\mu}}(\mathbf{a}_{d+1}) \geqslant \cdots \geqslant v_{\overrightarrow{\mu}}(\mathbf{a}_e) \end{align*} $$

and $ e = \ell m \text {.} $ In this way, we may denote elements of $[\ell ]^m$ as

$$ \begin{align*}\mathbf{a}_i := (a_{1i},\dots,a_{mi}) \text{,} \end{align*} $$

for $1 \leqslant i \leqslant e$ .

Recall the Harder and Narasimhan probabilities

$$ \begin{align*}p_i := \frac{r_i}{r} \text{,} \end{align*} $$

for $i = 1,\dots ,\ell $ . Then

$$ \begin{align*}\sum_{i=1}^\ell p_i = 1, \end{align*} $$

and, for later use in Section 7, we note that

(6.1) $$ \begin{align} \sum_{i = 1}^d \prod_{j=1}^m \frac{r_{a_{ji}}}{r} = \sum_{\mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m \frac{r_{a_j}}{r} = \sum_{\mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m p_{a_j} \text{.} \end{align} $$

Having fixed notation as above, our main theorem (Theorem 6.1) expands upon [Reference Codogni and Patakfalvi4, Theorem 5.11].

Theorem 6.1 Let $\operatorname {HN}\left (\overrightarrow {\mu }, \overrightarrow {r} \right )$ be Harder and Narasimhan data. Assume that $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ have positive degree. Let $\overrightarrow {p} = (p_1,\dots ,p_\ell )$ be its probability vector. Fix a positive integer $m>0$ and a nonnegative integer $z \geqslant 0$ . It then holds true that

$$ \begin{align*}\lim_{m \to \infty} \sum_{\mathbf{a} \in S_{m,z}(\overrightarrow{\mu}) } \prod_{j=1}^m p_{a_j} = 1 \text{.} \end{align*} $$

Proof Let

$$ \begin{align*}\mathcal{Y} = \mathcal{Y}\left([\ell],p_1,\dots,p_{\ell}\right)\end{align*} $$

be the discrete probability space on the set $[\ell ]$ and having probability measures $p_1,\dots ,p_\ell $ . Let

$$ \begin{align*}Y_j := Y_j(\overrightarrow{\mu})\end{align*} $$

be a sequence of independent and identically distributed random variables of $\mathcal {Y}$ that take value $\mu _i$ on i.

Let

$$ \begin{align*}Z_m := \sum_{j=1}^m Y_j \text{.} \end{align*} $$

Then, by considering the definition of the set $S_{m,z}(\overrightarrow {\mu })$ and the random variables $Y_1, Y_2,\dots $ , it follows that

(6.2) $$ \begin{align} \operatorname{Prob}\left( Z_m \geqslant z \right) = \operatorname{Prob}\left(\sum_{j=1}^m Y_j \geqslant z \right) = \sum_{\mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m p_{a_j} \text{.} \end{align} $$

Now, we apply the Central Limit Theorem (Theorem 2.3), to show that

(6.3) $$ \begin{align} \lim_{m \to \infty} \operatorname{Prob}\left( Z_m \geqslant z \right) = 1 \text{.} \end{align} $$

To this end, first note that $\mathcal {Y}$ is a finite metric space. Moreover, the independent and identically distributed random variables $Y_j$ have finite mean and variance. These quantities are independent of j. Denote them, respectively, by $\mu $ and $\sigma $ .

The Central Limit Theorem, for independent and identically distributed random variables (Theorem 2.3), thus implies that the random variable

$$ \begin{align*}\frac{Z_m - m \mu}{\sqrt{m}}\end{align*} $$

converges weakly to a normal distribution $\Phi $ with expected value $0$ and covariance $\sigma ^2$ .

In particular, for each real number y, it holds true that

$$ \begin{align*}\lim_{m\to \infty} \operatorname{Prob}\left( \frac{ Z_m - m \mu}{ \sqrt{m} } \geqslant y \right) = \operatorname{Prob}\left( \Phi \geqslant y \right) \text{.} \end{align*} $$

Now, recall that the Harder and Narasimhan data $\operatorname {HN}\left (\overrightarrow {\mu },\overrightarrow {r}\right )$ have positive degree. It thus follows that

$$ \begin{align*}\mu = \sum_{i=1}^\ell \mu_i p_i = \frac{\sum_{i=1}^\ell \mu_i r_i }{r}> 0 \text{.} \end{align*} $$

Thus, fixing a real number y, there is a positive integer $m_y> 0$ such that if $m \geqslant m_y\text {,}$ then

$$ \begin{align*}z \leqslant y \sqrt{m} + m \mu \leqslant Z_m \end{align*} $$

provided that

$$ \begin{align*}\frac{Z_m - m \mu}{\sqrt{m}} \geqslant y \text{.} \end{align*} $$

The above discussion implies, in particular, that

$$ \begin{align*}\liminf_{m \to \infty} \operatorname{Prob} \left(Z_m \geqslant z \right) \geqslant \operatorname{Prob} \left(\Phi \geqslant y \right) \end{align*} $$

for all real numbers y. On the other hand, note that

$$ \begin{align*}\lim_{y \to - \infty} \operatorname{Prob} \left(\Phi \geqslant y \right) = 1 \text{.} \end{align*} $$

Finally, since

$$ \begin{align*}\liminf_{m \to \infty} \operatorname{Prob}\left( Z_m \geqslant z \right) = 1 \end{align*} $$

and

$$ \begin{align*}\operatorname{Prob}\left( Z_m \geqslant z \right) \leqslant 1 \text{,} \end{align*} $$

for all m, it holds true that

$$ \begin{align*}\lim_{m \to \infty} \operatorname{Prob}\left( Z_m \geqslant z \right) = 1 \text{.} \end{align*} $$

This establishes the validity of the relation (6.3).

7 Proof of Theorem 1.5

Finally, we indicate the manner in which Theorem 1.5 follows from Theorems 1.4 and 6.1. The key point to the proof is to adapt the proof of [Reference Codogni and Patakfalvi4, Proposition 5.9] to the context of filtered vector spaces. Having done this, Theorem 1.5 follows from Theorem 6.1, essentially because of the relation (6.1).

Proof Consider the Harder and Narasimhan filtration (1.3) that is associated with the filtered vector space (1.2). Recall, respectively, the corresponding slopes, degrees, and ranks

$$ \begin{align*}\mu_i = \mu(V_i / V_{i-1}) \text{, } r_i = \dim (V_i / V_{i-1}), \text{ and } d_i = r_i \mu_i \text{,} \end{align*} $$

for $i = 1,\dots ,\ell $ . Put

$$ \begin{align*}r := \dim V \text{.} \end{align*} $$

For each

$$ \begin{align*}\mathbf{a} \in [\ell]^m \text{,} \end{align*} $$

recall that we have put

$$ \begin{align*}v_{\overrightarrow{\mu}}(\mathbf{a}) = \sum_{j=1}^m \mu_{a_j} \text{.} \end{align*} $$

Moreover, we arrange the elements of $[\ell ]^m$ in decreasing order with respect to $v_{\overrightarrow{\mu}}(\cdot )$ . We write this as

$$ \begin{align*}[\ell]^m := \{\mathbf{a}_1=(a_{11},\dots,a_{m1}),\dots,\mathbf{a}_{\ell m} = (a_{1 \ell m },\dots,a_{m \ell m}) \} \text{.} \end{align*} $$

Finally, we have defined the set

$$ \begin{align*}S_{m,z}(\overrightarrow{\mu}) := \left\{\mathbf{a} = (a_1,\dots,a_m) \in [\ell]^m : \sum_{j=1}^m \mu_{a_j} \geqslant z \right\} \text{,} \end{align*} $$

then $S_{m,z}(\overrightarrow {\mu })$ consists of the first $\# S_{m,z}(\overrightarrow {\mu })$ elements of $[\ell ]^m$ , with respect to the ordering that is induced by $v_{\overrightarrow {\mu}}(\cdot )$ .

Now, for all $1 \leqslant i \leqslant \ell m$ , inside of $V^{\otimes m}$ , define

$$ \begin{align*}F^i := \bigotimes_{j=1}^m V_{a_{ji}} \text{, } \end{align*} $$

$$ \begin{align*}H^i := \sum_{j=1}^i F^i, \end{align*} $$

and

$$ \begin{align*}G^i := \bigotimes_{j=1}^m \left(F^{a_{ji}} / F^{a_{ji} - 1} \right) \text{.} \end{align*} $$

Then

$$ \begin{align*}\dim(G^i) = \prod_{j=1}^m r_{a_{ji}}\text{.} \end{align*} $$

Let us also mention that, with respect to the filtration on $G^i$ that is induced by the given filtration on V, the slope of $G^i$ is

$$ \begin{align*}\mu(G^i) = \sum_{j=1}^m \mu(F^{a_{ji}} / F^{a_{ji}-1}) = \sum_{j=1}^m \mu_{a_{ji}} \text{.} \end{align*} $$

(We refer to [Reference Fujimori9, Section 1], for example, for more details about the behavior of slopes under taking tensor products and quotients.)

We next establish isomorphisms

(7.1) $$ \begin{align} H^i / H^{i-1} \simeq G^i \text{,} \end{align} $$

for $1 \leqslant i \leqslant \ell m$ . The isomorphisms (7.1) are a key point to the proof of Theorem 1.5. Assuming their existence, it then follows that

(7.2) $$ \begin{align} \dim H^{\# S_{m,z}(\overrightarrow{\mu})} = \sum_{i=1}^{\# S_{m,z}(\overrightarrow{\mu})} \dim G^i = \sum_{i=1}^{\# S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m r_{a_{ji}} \text{.} \end{align} $$

The desired conclusion then follows, since, in light of (7.2), it follows that

$$ \begin{align*}\frac{\dim H^{\# S_{m,z}(\overrightarrow{\mu})}}{\dim V^{\otimes m}} = \sum_{i=1}^{\# S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m \frac{r_{a_{ji}}}{r} \text{.} \end{align*} $$

However, by (6.1), this may be rewritten as

$$ \begin{align*}\frac{\dim H^{\# S_{m,z}(\overrightarrow{\mu})}}{\dim V^{\otimes m}} = \sum_{ \mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m \frac{r_{a_j}}{r} = \sum_{ \mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m p_{a_j} \text{.} \end{align*} $$

In particular, it then follows from Theorem 6.1 that

$$ \begin{align*}\lim_{m \to \infty} \frac{\dim H^{\# S_{m,z} (\overrightarrow{\mu})}}{ \dim V^{\otimes m} } = \lim_{m\to\infty} \sum_{\mathbf{a} \in S_{m,z}(\overrightarrow{\mu})} \prod_{j=1}^m p_{a_j} = 1 \text{.} \end{align*} $$

It remains to establish the isomorphisms (7.1), for each $1 \leqslant i \leqslant \ell m$ . To this end, we first make note the following vector space isomorphisms:

$$ \begin{align*}H^{i} / H^{i-1} = (H^{i-1} + F^i) / H^{i-1} \simeq F^i / \left(F^i \bigcap H^{i-1}\right) \end{align*} $$

and

$$ \begin{align*}G^i \simeq F^i / \left(\sum_{\mathbf{a}_{i} \succ \mathbf{a}_{i'}} F^{i'}\right) \text{;} \end{align*} $$

there is a naturally defined surjection

(7.3) $$ \begin{align} G^i \twoheadrightarrow H^i / H^{i-1} \text{.} \end{align} $$

However, on the other hand,

$$ \begin{align*}\dim(V^{\otimes m}) = \sum_{i=1}^{\ell m} \dim (H^i / H^{i-1}) \leqslant \sum_{i=1}^{\ell m} \dim G^i \end{align*} $$

and

$$ \begin{align*}\sum_{i=1}^{\ell m} \dim G^i = \sum_{\mathbf{a}_i = (a_{1i},\dots,a_{mi}) \in [\ell]^m} \left( \prod_{j=1}^m r_{a_{ji}}\right) = \left(\sum_{i=1}^\ell r_i \right)^m = \dim V^{\otimes m} \text{.} \end{align*} $$

The conclusion is then that

$$ \begin{align*}\dim(V^{\otimes m}) = \sum_{i =1}^{\ell m} \dim G^i \text{;} \end{align*} $$

whence, by dimension reasons,

$$ \begin{align*}\dim(H^i / H^{i-1}) = \dim G^i \text{,} \end{align*} $$

for all $1 \leqslant i \leqslant \ell m$ . The surjections (7.3) and thus injective. The desired isomorphisms (7.1) are thus established.