# Connectedness of the Free Uniform Spanning Forest as a function of edge
weights

Marcell Alexy  Márton Borbényi  András Imolay  Ádám Timár

###### Abstract

Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected
transitive graph $H$. It is shown in [3] that the Free Uniform Spanning Forest
($\mathsf{FSF}$) of this graph may not be connected, but the dependence of
this connectedness on $H$ remains somewhat mysterious. We study the case when
a positive weight $w$ is put on the edges of the $H$-copies in $G$, and
conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase
transition. For large enough $w$ we show that the $\mathsf{FSF}$ is connected,
while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is disconnected
when $w$ is small (relying on [3]). Finally, we prove that when $H$ is the
graph of one edge, then for any $w$, the $\mathsf{FSF}$ is a single tree, and
we give an explicit formula for the distribution of the distance between two
points within the tree.

##  1 Introduction

Consider some finite graph $H$ with a weight function (“conductances”)
$\hat{w}:E(H)\to{\mathbb{R}}^{+}_{0}$ on its edges. One may take an unweighted
graph and view it as one where the weights are constant 1. Choose a spanning
tree of $H$ at random, where the probability of a spanning tree $T$ will be
proportional to $\Pi_{e\in E(T)}\hat{w}(e)$. The so-defined probability
measure is called the Uniform Spanning Tree ($\mathsf{UST}$) of $(H,\hat{w})$.
For a given infinite graph $G$ and conductances $\hat{w}$, consider some
exhaustion of $G$ by a sequence of connected finite graphs $G_{n}$, and let
$\mathsf{UST}(G_{n})$ be the $\mathsf{UST}$ of the weighted graph
$(G_{n},\hat{w}|_{G_{n}})$. It is known that the weak limit of
$\mathsf{UST}(G_{n})$ exists, meaning that for any
$e_{1},...,e_{k},f_{1},...,f_{m}\in E(G)$,
$\mathbb{P}(e_{1},...,e_{k}\in\mathsf{UST}(G_{n}),\,f_{1},...,f_{m}\not\in\mathsf{UST}(G_{n}))$
converges, and to the same limit for any choice of the sequence $G_{n}$. The
limiting measure is called the Free Uniform Spanning Forest ($\mathsf{FUSF}$
or $\mathsf{FSF}$) of $(G,\hat{w})$. See [2] for background, references, and
the basic properties of the $\mathsf{FSF}$.

It was generally expected that the $\mathsf{FSF}$ of “tree-like graphs” would
consist of a single tree, until Gábor Pete and the last author showed in [3]
that for suitably chosen $d$ and connected finite transitive graph $H$, the
Cartesian product $\mathbb{T}^{d}\square H$ of the $d$-regular tree
$\mathbb{T}^{d}$ and $H$ has a disconnected $\mathsf{FSF}$. From the proof,
however, it is not clear what happens to the disconnectedness of the
$\mathsf{FSF}$ if we do some natural changes to $d$ and $H$, e.g., increase
$d$ with a fixed $H$, or fix $d$ and take a lift of $H$. No monotonicity
result of this type is known. A question in the same spirit is to ask how the
connectedness of the $\mathsf{FSF}$ changes if we put constant positive weight
$w$ on every edge of the $H$-copies in $\mathbb{T}^{d}\square H$ and then
change $w$. What happens if $w$ is very small or large? Does there always
exist some $w$ where there is a single $\mathsf{FSF}$ component? Is there
always some $w$ where there are infinitely many $\mathsf{FSF}$ components
(with an $H$ that has at least 2 vertices)? Is there any kind of monotonicity
in $w$, and perhaps even a critical value that separates the phases of
disconnectedness and connectedness? The present paper contributes to the
understanding of these questions. In particular, initial steps are taken in
Conjecture 1.1.

Let $H$ be a finite connected graph and $\mathbb{T}^{d}$ be the $d$-regular
tree. For an arbitrary given $w>0$, define the weight function $\hat{w}$ on
the edges of $G=\mathbb{T}^{d}\square H$ so that $\hat{w}(e)=1$ if $e$ is of
the form $\\{(x_{1},y),(x_{2},y)\\}$ and $\hat{w}(e)=w$ if $e$ is of the form
$\\{(x,y_{1}),(x,y_{2})\\}$. Define $\mathsf{FSF}_{w}(G)$ as the
$\mathsf{FSF}$ of $(G,\hat{w})$.

######  Conjecture 1.1.

If $\mathsf{FSF}_{w^{\prime}}$ and $\mathsf{FSF}_{w^{\prime\prime}}$ are
connected for some $w^{\prime\prime}>w^{\prime}>0$ then $\mathsf{FSF}_{w}$ is
connected for every $w\in[w^{\prime},w^{\prime\prime}]$. Similar statement
holds for disconnectedness. Moreover, there exists a $\gamma\in[0,\infty]$
such that $\mathsf{FSF}_{w}$ has a unique component whenever $w>\gamma$, and
$\mathsf{FSF}_{w}$ has infinitely many components whenever $w<\gamma$.

We mention that having more than one component automatically implies having
infinitely many for a much wider class of transitive graphs than the ones
considered here ([1], [5], or see [3] for a short direct proof for the special
product graphs that we consider here).

The simplest nontrivial example of a graph of the form $\mathbb{T}^{d}\square
H$ is the case when $H=K_{2}$ is a single edge. It is not clear what to
expect: on one hand the graph may be “too close” to the tree to produce
disconnected FSF, on the other hand one may speculate that for small enough
$w$ the relatively large degree of the tree could be the reason for a similar
phenomenon as in [3] and make the FSF fall apart. Pengfei Tang has shown in
[4] that the $\mathsf{FSF}$ is connected for the unweighted question for
$\mathbb{T}^{d}\square H$. (His proof was worked out for a slightly different
graph, but it is mentioned in [4] that a similar argument can be applied for
$G=\mathbb{T}^{d}\square K_{2}$.) However, his method does not give a
quantitative insight into connectivity within the $\mathsf{FSF}$ of $G$, and
it does not seem to apply to the case when weights are added to the edges. The
method of [3], which needs $H$ to be relatively large, did not give an insight
into this special case either. We settle this question of $H=K_{2}$ through an
enumeration, which will also enable us to bound the decay of the distance
between two points (see Lemma 2.10).

######  Theorem 1.2.

For every $w>0$, $\mathsf{FSF}_{w}(\mathbb{T}^{d}\square K_{2})$ is connected.

We verify Conjecture 1.1 for large enough $w$, with only the assumption that
$H$ is regular, finite and connected. Furthermore, we roughly sketch how the
arguments in [3] can be applied to show that the conjecture holds in a
neighborhood of 0 for a large class of $\mathbb{T}^{d}$ and $H$, as in the
next theorem.

######  Theorem 1.3.

For every finite connected regular graph $H$ and $d$-regular tree
$\mathbb{T}^{d}$, there exists a $W<\infty$ such that for every $w>W$ the
forest $\mathsf{FSF}_{w}(\mathbb{T}^{d}\square H)$ is connected. Conversely,
if $H$ is transitive, $d$ is large enough compared to the degree in $H$, and
$|H|>d^{5/2}$, then for every $w\leq 1$ the forest
$\mathsf{FSF}_{w}(\mathbb{T}^{d}\square H)$ has infinitely many components
almost surely.

###  1.1 Notation

Denote by $t(G)$ the (weighted) number of spanning trees of a finite graph
$G$. Let $T\square wH$ be the graph obtained by the Cartesian product of $T$
and $H$, with $w$ weights on the edges of the form
$\\{(x,y_{1}),(x,y_{2})\\}$, and weight $1$ on the rest of the edges. For
shorthand throughout Section 2 we use $\hat{T}=T\square wK_{2}$ for any graph
$T$. In a graph product $T\square wH$ we call bag the subgraphs of the form
$\\{v\\}\times wH$ where $v\in T$. $T_{n}$ will denote a ball with radius $n$
around a fix vertex $u$ in $\mathbb{T}^{d}$. With a (convenient) slight
redundancy, the $\mathsf{FSF}$ of $T\square wH$ is the same object as the
$\mathsf{FSF}_{w}$ of $T\square H$.

We will rely on one particular consequence of Wilson’s algorithm on finite
graphs [6], namely, that for a finite connected graph $G$, the path between
points $a,b\in V(G)$ within $\mathsf{UST}(G)$ has the same distribution as the
loop-erased random walk path $\text{LERW}_{G}(a\to b)$ from $a$ to $b$, which
is constructed as follows. Run random walk in $G$ starting from $a$ until
hitting $b$, and erase all the loops in the order of their appearence along
the walk, to obtain a simple path from $a$ to $b$. The same link between the
$\mathsf{UST}$ and LERW is true when $G$ has positive edge weights, in which
case random walk on this network is understood instead of simple random walk,
with weights being the conductances. See Chapter 4.1 of [2] for more details.
In general, for an arbitrary walk $(X_{1},...,X_{n})$, LE$(X_{1},...,X_{n})$
will denote its loop-erasure.

##  2 Product of a tree and a weighted edge

###  2.1 Recursive formulas for the number of weighted spanning trees

Fix a constant $d\geq 3$. Most of the definitions in this section depend on
$d$, but we usually will not write it as an index.

Let $u\in\mathbb{T}^{d}$ a fixed vertex, remember that $T_{n}$ is the ball
around $u$ with radius $n$.

######  Definition 2.1.

Define the perfect $(d-1)$-ary tree with height $n$ recursively in the
following way. A perfect $(d-1)$-ary tree with height $0$ is a single vertex,
the root. For $n>0$ a perfect $(d-1)$-ary tree with height $n$ has a root, and
it is connected with the roots of $d-1$ pieces of perfect $(d-1)$-ary trees
with height $n-1$.

For brevity, from now on we call the perfect $(d-1)$-ary tree simply as a
perfect tree and we denote the height $n$ perfect tree by $A_{n}$.

o

$A_{2}$ with $d=4$

u

$T_{2}$ with $d=4$

An alternative way to define the perfect tree is that if we delete an edge
incident with $u$ from $T_{n}$, then the component containing $u$ is $A_{n}$
(and the other component is an $A_{n-1}$).

Let $o$ be the root of $A_{n}$. Denote by
$A_{n-1}^{1},A_{n-1}^{2},...,A_{n-1}^{d-1}$ the subgraphs of $A_{n}$ from the
recursive Definition 2.1, and let their roots be $o_{1},o_{2},...,o_{d-1}$, so
these are the neighbours of $o$ in $A_{n}$. Let $e$ be the edge between
$(o,0)$ and $(o,1)$ in $\hat{A}_{n}$. In $\hat{A}_{n}$ and $\hat{T}_{n}$ let
$(v,0)$ and $(v,1)$ be the two vertices in the bag $\\{v\\}\square wK_{2}$ for
any vertex $v$. Let $G_{i}$ be the subgraph of $\hat{A}_{n}$ spanned by the
vertices of $\hat{A}_{n-1}^{i}$ and $(o,0)$ and $(o,1)$ for all $1\leq i\leq
d-1$. Note that each edge of $\hat{A}_{n}$ is exactly in one of the $G_{i}$’s
except $e$, which is contained in all of the $G_{i}$’s.

######  Definition 2.2.

For a shorthand of $t(\hat{A}_{n})$ we use $a_{n}$, and let $a^{\prime}_{n}$
denote the weighted number of spanning trees of $\hat{A}_{n}$ containing $e$.

We prove recursive formulas for these quantities.

######  Lemma 2.3.

$a^{\prime}_{n}=w\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d-1}$

$\hat{A}_{2}$$\hat{A}^{1}_{1}$$e$$...$$...$

One of the edges from $\\{(o,0),(o_{1},0)\\}$
and $\\{(o,1),(o_{1},1)\\}$ is in $T$.
$T\cap\hat{A}^{1}_{1}$ is a spanning tree in $\hat{A}^{1}_{1}$.

$\hat{A}_{2}$$\hat{A}^{1}_{1}$$e$$e_{1}$$...$$...$

Both edges from $\\{(o,0),(o_{1},0)\\}$
and $\\{(o,1),(o_{1},1)\\}$ are in $T$.
$T\cap\hat{A}^{1}_{1}\cup\\{e_{1}\\}$ is a spanning tree in $\hat{A^{1}_{1}}$

###### Proof.

The $w$ multiplier comes from the weight $w$ on $e$.

Note that a subgraph $T$ of $\hat{A}_{n}$ is a spanning tree containing $e$ if
and only if $T\cap G_{i}$ is a spanning tree containing $e$ for all $1\leq
i\leq d-1$.

So we need to count the weighted number of spanning trees of $G_{i}$
containing $e$ (not multiplying with the weight $w$ on $e$ as we already
counted the weight of $e$). Let $T$ be such a spanning tree. Consider the
edges $\\{(o,0),(o_{i},0)\\}$ and $\\{(o,1),(o_{i},1)\\}$. This is a cut of
$G_{i}$, so at least one of them must be in $T$. If exactly one, then we have
$2$ options choosing which one, and $T\cap\hat{A}^{i}_{n-1}$ must be a
spanning tree of $\hat{A}^{i}_{n-1}$, so there are $2a_{n-1}$ weighted
options. If both of the edges are in $T$, then
$T\cap\hat{A}^{i}_{n-1}\cup\\{\\{(o_{i},0),(o_{i},1)\\}\\}$ is a spanning tree
of $\hat{A}^{i}_{n-1}$, and any spanning tree of $\hat{A}^{i}_{n-1}$
containing the edge $\\{(o_{i},0),(o_{i},1)\\}$ minus the edge
$\\{(o_{i},0),(o_{i},1)\\}$ does arise as $T\cap\hat{A}^{i}_{n-1}$, so this is
bijection, and it gets a $\frac{1}{w}$ multiplier when we count the weighted
number, as the weight of $\\{(o_{i},0),(o_{i},1)\\}$ is $w$. So this is
$\frac{1}{w}a^{\prime}_{n-1}$ weighted options. Therefore, independently for
each $G_{i}$, we have $\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)$
weighted possibilities. The conclusion follows.

∎

######  Lemma 2.4.

$a_{n}=a^{\prime}_{n}+(d-1)a_{n-1}\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d-2}$

$\hat{A}_{2}$$\hat{A}^{1}_{1}$$e$$...$$...$

Both edges from $\\{(o,0),(o_{1},0)\\}$ and
$\\{(o,1),(o_{1},1)\\}$ are in $T$, but $e$ is not.
$T\cap\hat{A}^{1}_{1}$ is a spanning tree in $\hat{A}^{1}_{1}$.

###### Proof.

The weighted number of spanning trees that contain $e$ is $a^{\prime}_{n}$. It
is easy to see that $T$ is a spanning tree that does not contain $e$, if and
only if $T\cap G_{i}$ is a spanning tree of $G_{i}$ not containing $e$ for
some $i$, and $T\cap G_{j}$ is a graph not containing $e$, with $T\cap
G_{j}\cup\\{e\\}$ is a spanning tree of $G_{j}$ for all $j\neq i$. We have
$d-1$ options to choose $i$, then $\\{(o,0),(o_{i},0)\\}$ and
$\\{(o,1),(o_{i},1)\\}$ must be in $T$ and $T\cap\hat{A}^{i}_{n-1}$ is a
spanning tree of $\hat{A}^{i}_{n-1}$, so this is $a_{n-1}$ weighted options.
For the other $j$’s it is exactly the same as in the proof of Lemma 2.3, so
$\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)$ weighted possibilities,
from which the proof is complete. ∎

######  Lemma 2.5.

$t(\hat{T}_{n})=2a_{n}a_{n-1}+\frac{1}{w}(a_{n}a^{\prime}_{n-1}+a^{\prime}_{n}a_{n-1})$

$\hat{A_{0}}$$\hat{A_{1}}$

One edge is in T,
$T\cap\hat{A}_{1}$ is a tree,
$T\cap\hat{A}_{0}$ is a tree.

$e_{0}$$\hat{A_{0}}$$\hat{A_{1}}$

Both edges are in T,
$T\cap\hat{A}_{1}$ is a tree,
$T\cap\hat{A}_{0}\cup\\{e_{0}\\}$ is a tree.

$e_{1}$$\hat{A_{0}}$$\hat{A_{1}}$

Both edges are in T,
$T\cap\hat{A}_{1}\cup\\{e_{1}\\}$ is a tree,
$T\cap\hat{A}_{0}$ is a tree.

###### Proof.

$T_{n}$ can be constructed by taking an $A_{n}$ graph and an $A_{n-1}$ graph
and connecting their roots. In each spanning tree $T$ of $\hat{T}_{n}$ either
$T\cap\hat{A}_{n}$ or $T\cap\hat{A}_{n-1}$ is a spanning tree, or both. If
both, then we have $2$ options to connect them, so it is $2a_{n}a_{n-1}$
weighted options. If $\hat{A}_{n}\cap T$ is a spanning tree, but
$\hat{A}_{n-1}\cap T$ is disconnected, then we have to put both edges between
$\hat{A}_{n}$ and $\hat{A}_{n-1}$ into $T$, and as in Lemma 2.3, we can think
of $\hat{A}_{n-1}\cap T$ as a spanning tree containing the edge between the
$2$ vertices in the bag of the root of $\hat{A}_{n-1}$, minus this edge, so it
is $\frac{1}{w}a_{n}a^{\prime}_{n-1}$ weighted possibilities. In the same way
we get $\frac{1}{w}a^{\prime}_{n}a_{n-1}$ for the third case. Summing these we
get the desired result. ∎

Let $t_{m}(\hat{T}_{n})$ be the number of spanning trees in $\hat{T}_{n}$ with
the unique path from $(u,0)$ to $(u,1)$ going into exactly $m$ bags. Note that
as a bag only contains $2$ vertices, we don’t have plenty of options for a
path between $(u,0)$ and $(u,1)$. The only way for a path through $m$ bags is
that we do $m-1$ moves in tree edges, going into the $m$’th bag, then in the
$m$’th step we move within the bag, and then $m-1$ steps back up in tree
edges.

######  Lemma 2.6.

For each pair $n>m\geq 2$ we have

$t_{m}(\hat{T}_{n})=d(d-1)^{m-2}w\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)\left(2a_{n-m}+\frac{1}{w}a^{\prime}_{n-m}\right)\prod_{i=1}^{m}\left(2a_{n-i}+\frac{1}{w}a^{\prime}_{n-i}\right)^{d-2},$

and in the $m=1$ case, for $n>1$ the following is true.

$t_{1}(\hat{T}_{n})=w\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d}.$

…………$\hat{A}_{2}$$\hat{A}_{2}$$\hat{A}_{1}$$\hat{A}_{0}$$\hat{A}_{0}$

A possible spanning tree $T$ of $\hat{T}_{3}$ with the path highlighted
between the vertices in the root bag. $2$ pieces of $\hat{A}_{2}$, $1$ piece
of $\hat{A}_{1}$, and $2$ pieces $\hat{A}_{0}$ are hanging from the main path.

###### Proof.

The proof of the $m=1$ case is the same as the proof of Lemma 2.3, except here
the root has degree $d$, so the exponent is $d$ instead of $d-1$.

For the $m>1$ case there are $d(d-1)^{m-2}$ paths from $(u,0)$ to $(u,1)$
touching $m$ bags, as the first $m-1$ steps determine the path, and this is an
arbitrary $m-1$ long path in $T_{n}$, so for the first step we have $d$
options, and for the rest there are $d-1$ possibilities. We always have
exactly $1$ step within a bag in the path, which gives the multiplier $w$ in
the equation.

Now assume that we know the path from $(u,0)$ to $(u,1)$. We want to count the
number of spanning trees containing this path. The projection of this path to
$T_{n}$ is a path of length $m-1$, with $u$ as one of its endpoints. No matter
what this path is, there are always $d-1$ pieces of $A_{n-1}$, $d-2$ pieces of
$A_{n-2}$, $d-2$ pieces of $A_{n-3}$, …, $d-2$ pieces of $A_{n-m+1}$ and $d-1$
pieces of $A_{n-m}$ subtrees, that are hanging from the path, i.e. disjoint
from the projected path, and with root connected to it. As in the proof of
Lemma 2.3 we have independently $(2a_{n-i}+\frac{1}{w}a^{\prime}_{n-i})$
weighted possibilities for each $\hat{A}_{n-i}$ so that the whole subgraph is
a spanning tree. Multiplying these we get the number of spanning trees with
this path. ∎

###  2.2 Distribution of the distances in
$\mathsf{FSF}_{w}(\mathbb{T}^{d}\square K_{2})$

Let $A$ be the infinite tree with degrees $d$, except one vertex, which has
degree $d-1$, call this special vertex $o$. Let $e$ be the edge in the bag of
$o$ in $\hat{A}$. Define $c:=\mathbb{P}(e\in\mathsf{FSF}_{w}(A\square
K_{2}))$. The $\hat{A}_{n}$ is an exhaustion of $\hat{A}$, so by the
definition of the $\mathsf{FSF}_{w}$ we have

$c=\lim_{n\to\infty}\frac{a^{\prime}_{n}}{a_{n}}.$

Let $c_{n}=\frac{a^{\prime}_{n}}{a_{n}}.$

######  Lemma 2.7.

The sequence $s_{n}:=\lim_{n\to\infty}\frac{a_{n-1}^{d-1}}{a_{n}}$ converges
to a number $s$ and

$s=\frac{c}{w\big{(}2+\frac{c}{w}\big{)}^{d-1}}.$

###### Proof.

From Lemma 2.3 we have

$c_{n}a_{n}=w\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d-1}=wa_{n-1}^{d-1}\Big{(}2+\frac{c_{n-1}}{w}\Big{)}^{d-1}.$

After rearranging and letting $n\to\infty$,

$\frac{c}{w\big{(}2+\frac{c}{w}\big{)}^{d-1}}=\lim_{n\to\infty}\frac{c_{n}}{w\left(2+\frac{c_{n-1}}{w}\right)^{d-1}}=\lim_{n\to\infty}\frac{a_{n-1}^{d-1}}{a_{n}}=s.$

∎

From Lemma 2.4,

$a_{n}=a^{\prime}_{n}+(d-1)a_{n-1}\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d-2}=c_{n}a_{n}+(d-1)a_{n}s_{n}\left(2+\frac{c_{n-1}}{w}\right)^{d-2}.$

Dividing this by $a_{n}$, taking $n\rightarrow\infty$ and substituting the
identity from Lemma 2.7. we get

$1=c+\frac{(d-1)c\big{(}2+\frac{c}{w}\big{)}^{d-2}}{w\big{(}2+\frac{c}{w}\big{)}^{d-1}}=c+\frac{c(d-1)}{2w+c}.$

After rearranging we get a quadratic equation of $c$:

$c^{2}+c(2w+d-2)-2w=0$ (2.1)

The constant term is negative, so we have two real roots, a negative and a
positive, and $c>0$ so we get the following Theorem.

######  Theorem 2.8.

$\mathbb{P}(e\in\text{$\mathsf{FSF}_{w}$}(A\square
K_{2}))=c=\frac{2-d-2w+\sqrt{(2w+d-2)^{2}+8w}}{2}$

###### Proof.

From (2.1) with quadratic formula. ∎

######  Remark 2.9.

We can also get a formula for $s$ if we substitute the equation from Theorem
2.8 to the equation in Lemma 2.7.

Let $q_{m}:=\lim_{n\to\infty}\frac{t_{m}(\hat{T}_{n})}{t(\hat{T}_{n})}$. This
number has another meaning. This is the probability that for a
$u\in\mathbb{T}^{d}$, $(u,0)$ and $(u,1)$ belong to the same component of
$\mathsf{FSF}_{w}$ in $\mathbb{T}^{d}\square K_{2}$, and their distance in the
tree is $2m-1$, in other words, the path between them uses $m$ bags.

######  Lemma 2.10.

For any integer $m\geq 2$, $q_{m}=K\left(\frac{(d-1)c}{2w+c}\right)^{m}$,
where $K=\frac{d(2w+c)^{2}}{(2w+2c)(d-1)^{2}}$ is a constant that does not
depend on $m$.

###### Proof.

We call two positive sequences ($u_{n},\ v_{n}$) equivalent ($u_{n}\sim
v_{n}$) if $\lim_{n\to\infty}u_{n}/v_{n}=1$.

We are going to prove, that $t_{m}(\hat{T}_{n})$ and $t(\hat{T}_{n})$ are
asymptotically the same as a constant times $a_{n}a_{n-1}$.

$\lim_{n\to\infty}\frac{t_{m}(\hat{T}_{n})}{t(\hat{T}_{n})}=\lim_{n\to\infty}\frac{a_{n}a_{n-1}}{t(\hat{T}_{n})}\lim_{n\to\infty}\frac{t_{m}(\hat{T}_{n})}{a_{n}a_{n-1}}.$

Recall the following constants:

$\lim_{n\to\infty}\frac{a^{\prime}_{n}}{a_{n}}=c\ \ \text{and}\ \
\lim_{n\to\infty}\frac{a_{n-1}^{d-1}}{a_{n}}=s.$

Using these, and Lemma 2.5,

$\lim_{n\to\infty}\frac{a_{n}a_{n-1}}{t(\hat{T}_{n})}=\lim_{n\to\infty}\frac{a_{n}a_{n-1}}{2a_{n}a_{n-1}+\frac{1}{w}(a_{n}a^{\prime}_{n-1}+a^{\prime}_{n}a_{n-1})}=\lim_{n\to\infty}\frac{1}{2+\frac{1}{w}(\frac{a^{\prime}_{n-1}}{a_{n-1}}+\frac{a^{\prime}_{n}}{a_{n}})}=\frac{1}{2+\frac{2c}{w}}.$

From Lemma 2.6,

$\displaystyle\lim_{n\to\infty}\frac{t_{m}(\hat{T}_{n})}{a_{n}a_{n-1}}=$
$\displaystyle\lim_{n\to\infty}\frac{d(d-1)^{m-2}w\Big{(}2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\Big{)}\Big{(}2a_{n-m}+\frac{1}{w}a^{\prime}_{n-m}\Big{)}\prod_{i=1}^{m}\Big{(}2a_{n-i}+\frac{1}{w}a^{\prime}_{n-i}\Big{)}^{d-2}}{a_{n}a_{n-1}}.$

Here $\big{(}2a_{n-i}+\frac{1}{w}a^{\prime}_{n-i}\big{)}\sim
a_{n-i}\big{(}2+\frac{c}{w}\big{)}$. Hence

$t_{m}(\hat{T}_{n})\sim
wd(d-1)^{m-2}a_{n-1}\left(\prod_{i=1}^{m}a_{n-i}^{d-2}\right)a_{n-m}\Big{(}2+\frac{c}{w}\Big{)}^{(d-2)m+2}.$

Using the fact, that $a^{d-1}_{n-i}\sim sa_{n-i+1}$, one can prove easily by
induction, that $a_{n-1}\left(\prod_{i=1}^{m}a_{n-i}^{d-2}\right)a_{n-m}\sim
a_{n-1}a_{n}s^{m}$. Thus, combining the two calculations,

$\frac{t_{m}(\hat{T}_{n})}{a_{n}a_{n-1}}\sim
wd(d-1)^{m-2}\Big{(}2+\frac{c}{w}\Big{)}^{(d-2)m+2}s^{m}.$

Therefore

$\lim_{n\to\infty}\frac{t_{m}(\hat{T}_{n})}{t(\hat{T}_{n})}=K(d-1)^{m}\Big{(}2+\frac{c}{w}\Big{)}^{(d-2)m}s^{m},$

where

$K=\frac{1}{2+\frac{2c}{w}}\cdot
wd\frac{1}{(d-1)^{2}}\left(2+\frac{c}{w}\right)^{2}=\frac{d(2w+c)^{2}}{(2w+2c)(d-1)^{2}}.$

It means, that $q_{m}=K\left((d-1)s(2+\frac{c}{w})^{d-2}\right)^{m}.$ Using
Lemma 2.7, we know, that $s=\frac{c}{w\left(2+\frac{c}{w}\right)^{d-1}}$, thus
$q_{m}=K\left(\frac{(d-1)c}{2w+c}\right)^{m}.$ ∎

######  Lemma 2.11.

$\mathbb{P}(\\{(u,0),(u,1)\\}\in\mathsf{FSF}_{w}(\mathbb{T}^{d}\square
K_{2}))=q_{1}=\frac{(2w+c)c}{2w+2c}.$

###### Proof.

From the $m=1$ case of Lemma 2.6, we have

$\lim_{n\to\infty}\frac{t_{1}(\hat{T}_{n})}{a_{n}a_{n-1}}=\lim_{n\to\infty}\frac{w\left(2a_{n-1}+\frac{1}{w}a^{\prime}_{n-1}\right)^{d}}{a_{n}a_{n-1}}=\lim_{n\to\infty}\frac{ws_{n}a_{n}a_{n-1}\left(2+\frac{c_{n-1}}{w}\right)^{d}}{a_{n}a_{n-1}}=ws\left(2+\frac{c}{w}\right)^{d}.$

In the proof of Lemma 2.10, we calculated

$\lim_{n\to\infty}\frac{a_{n}a_{n-1}}{t(\hat{T}_{n})}=\frac{1}{2+\frac{2c}{w}}.$

Combining these and using Lemma 2.7, we have

$q_{1}=\lim_{n\to\infty}\frac{t_{1}(\hat{T}_{n})}{t(\hat{T}_{n})}=\lim_{n\to\infty}\frac{a_{n}a_{n-1}}{t(\hat{T}_{n})}\lim_{n\to\infty}\frac{t_{1}(\hat{T}_{n})}{a_{n}a_{n-1}}=\frac{1}{2+\frac{2c}{w}}\cdot
c\left(2+\frac{c}{w}\right)=\frac{(2w+c)c}{2w+2c}.$

∎

Lemma 2.10 gives us that the distance of two vertices in the same bag in the
$\mathsf{FSF}_{w}$ has a geometric distribution.

######  Lemma 2.12.

Let $u\in\mathbb{T}^{d}$. Then $(u,0)$ and $(u,1)$ are in the same component
of the $\mathsf{FSF}_{w}$ of $\mathbb{T}^{d}\square K_{2}$ with probability
$1$.

###### Proof.

We want to prove that the path between $(u,0)$ and $(u,1)$ is almost surely
finite, so $\sum_{m=1}^{\infty}q_{m}=1$. From (2.1) we have
$2wc+c^{2}=2w+2c-cd$, hence

$\frac{(2w+c)c}{2w+2c}=\frac{2w+2c-cd}{2w+2c}=1-\frac{cd}{2w+2c},$

and (2.1) can also be transformed to $2w+c=\frac{2w+c-(d-1)c}{c}$, thus

$K=\frac{d(2w+c)^{2}}{(2w+2c)(d-1)^{2}}=\frac{d((2w+c)^{2}-(d-1)c(2w+c))}{(2w+2c)(d-1)^{2}c}=\frac{cd}{2w+2c}\cdot\frac{1-\frac{(d-1)c}{2w+c}}{\left(\frac{(d-1)c}{2w+c}\right)^{2}}.$

Combining these with Lemma 2.10 and Lemma 2.11, we have

$\sum_{m=1}^{\infty}q_{m}=\frac{(2w+c)c}{2w+2c}+K\sum_{m=2}^{\infty}\left(\frac{(d-1)c}{2w+c}\right)^{m}=1-\frac{cd}{2w+2c}+K\cdot\frac{\left(\frac{(d-1)c}{2w+c}\right)^{2}}{1-\frac{(d-1)c}{2w+c}}=1.$

∎

######  Lemma 2.13.

Let $H$ be an arbitrary finite connected graph and consider the weighted graph
$G=\mathbb{T}^{d}\square wH$. If any pair $a,b$ of vertices in the same bag
belong to the same component of $\mathsf{FSF}_{w}$ almost surely, then the
$\mathsf{FSF}_{w}$ is almost surely connected.

###### Proof.

Take two adjacent bags. Let the set of edges between them called $E^{\prime}$.
The event that at least one edge of $E^{\prime}$ is in $\mathsf{FSF}_{w}$ is a
cylinder event, for every graph $G_{n}$ of an exhausting finite sequence for
$G$ this event has probability one to hold for the $\mathsf{UST}$. Hence in
the $\mathsf{FSF}_{w}$ there is an edge from $E^{\prime}$ with probability
one. Thus there are always two connected vertices in the two adjacent bags. By
assumption, all vertices within a bag are in the same component, therefore all
vertices in these two adjacent bags are in the same component. This is true
for any two adjacent bags, thus for all edges in $\mathbb{T}^{d}$. Using
countable intersection, we conclude that the $\mathsf{FSF}_{w}$ of that graph
is connected with probability one. ∎

Now we have everything, to prove the main result of this section:

###### Proof of Theorem 1.2..

From Lemma 2.12. and Lemma 2.13. the statement follows. ∎

######  Remark 2.14.

It is a natural question to ask whether this method can be generalized to
other graphs instead of $K_{2}$. Unfortunately we strongly relied on the fact,
that in $\hat{T}_{n}$ a path between $(u,0)$ and $(u,1)$ looks quite nice,
while if we change $K_{2}$ to some larger graph then plenty of other options
arise which we cannot handle with this enumerative method.

##  3 The general case, large and small weights

In this section we are going to prove Theorem 1.3. The first part will follow
from the next theorem.

######  Theorem 3.1.

Given an arbitrary $d>2$ and a finite, regular, connected graph $H$, there is
a $W>0$ such that the $\mathsf{FSF}_{w}$ of the graph $G=\mathbb{T}^{d}\square
H$ is almost surely connected for all $w>W$.

As before, denote by $T_{n}$ the ball of radius $n$ in $\mathbb{T}^{d}$. Let
$U$ be the central bag of $T_{n}\square wH$: the bag that corresponds to the
center of this ball.

######  Definition 3.2.

A trip is a walk $(X_{1},X_{2},...,X_{T})$ such that $X_{i}\in U$ if and only
if $i=1$ or $T$.

######  Definition 3.3.

Bag $D$ is memorable for a trip $(X_{1},...,X_{T})$, if the trip intersects
$D$, and satisfies the following. If $\tau\in[1,T]$ is the last step when
$X_{\tau}\in D$, then for every bag $D^{\prime}$($\not=U,D$) that separates
$U$ and $D$, $V(D^{\prime})\not=V(D^{\prime})\cap\\{X_{\tau+1},...,X_{T}\\}$.

######  Proposition 3.4.

Let ${\cal X}=(X_{1},...,X_{T})$ be a walk in $T_{n}\square wH$, with $X_{1}$
in the central bag $U$ of $T_{n}\square wH$. Suppose that ${\cal
X}^{\prime}=(X_{k},X_{k+1},...,X_{k^{\prime}})$ is some subwalk which is a
trip and intersects bag $D$. Assume that $D$ is not memorable for ${\cal
X}^{\prime}$. Then the loop-erasure of $(X_{1},...,X_{k^{\prime}})$ does not
intersect $D$.

###### Proof.

By our assumptions there exists a largest number $\tau$ with
$k<\tau<k^{\prime}$ and $X_{\tau}\in D$. Since $D$ is not memorable, there
exists a bag $D^{\prime}$ that separates $U$ and $D$, and with the property
that $V(D^{\prime})=V(D^{\prime})\cap\\{X_{\tau+1},...,X_{k^{\prime}}\\}$. Let
$t$ be the first time after $\tau$ that we enter $D^{\prime}$. Such a $t$
exists, because $X_{\tau}\in D$ and $X_{k^{\prime}}\in U$. Let ${\cal L}$ be
LE$(X_{1},...,X_{t})$. If ${\cal L}\cap D=\emptyset$, then the claim is
proved, because we do not visit $D$ after $t>\tau$. Otherwise the first time
that ${\cal L}$ enters $D^{\prime}$ is strictly before $t$ (since ${\cal L}$
has to enter $D^{\prime}$ before entering $D$ and reentering $D^{\prime}$ at
$X_{t}$). Let this first vertex of entrance be $v$. By assumption on ${\cal
X}^{\prime}$, $(X_{t},...,X_{k^{\prime}})$ visits every vertex of
$D^{\prime}$. Let $t^{\prime}\geq t$ be the first time that
$X_{t^{\prime}}=v$. Then the loop-erasure of $(X_{1},...,X_{t^{\prime}})$
erases everything that happened after the first entrance to $D^{\prime}$ at
$v$. In particular, it erases every step in $D$ before $t^{\prime}$, so
LE$(X_{1},...,X_{t^{\prime}})\cap D=\emptyset$. Since $k^{\prime}>t^{\prime}$
(${\cal X}^{\prime}$ is a trip), and no step after $t^{\prime}>\tau$ is in
$D$, the claim is proved. ∎

######  Lemma 3.5.

There exists a $W>0$ such that for any $\alpha>0$ there is an $m$ such that
the following holds for every $w>W$. Let $u\in U$, $n>m$ and ${\cal
X}=(X_{1},...,X_{T})$ be a trip in $T_{n}\square wH$ with $X_{1}=u$. Then we
have

$\mathbb{P}({\cal X}\text{ has a memorable bag outside }T_{m}\square
wH)<\alpha.$

###### Proof.

Choose $\varepsilon:=\frac{1}{2(d-1)}$. Let $w>W$, where we specify $W$ at the
end of this paragraph. If $(Z_{1},Z_{2},...)$ is a random walk in
$\mathbb{T}^{d}\square wH$ started from a bag $B$, then let $\lambda$ be the
first time when it exits $B$. If $w$ was large enough, we have for any
starting vertex $x=Z_{1}$ and last vertex $y=Z_{\lambda}$

$\mathbb{P}(\\{Z_{1},...,Z_{\lambda}\\}=B\mid\ Z_{1}=x,\
Z_{\lambda}=y)>1-\varepsilon,$

because the minimum over $x$ and $y$ of the probability on the left tends to
$1$ as $w$ goes to infinity. Fix $W$ so that the above inequality holds.

Fix bag $D$; we will use notation from Definition 3.3. Let $t$ be the first
time after $\tau$ that we enter $D^{\prime}$. Let $A_{x}$ be the event that
the random walk started from a point $x\in D^{\prime}$ visits every vertex of
$D^{\prime}$ before leaving $D^{\prime}$. Denote by $\mathbb{P}_{x}$ the
distribution of a random walk ${\cal Y}=(Y_{1},...,Y_{R})$ started from
$x=Y_{1}$ and stopped at the first entrance to $U$. Let $B_{x}$ be the event
that ${\cal Y}$ does not visit $D$. Then

$\mathbb{P}((X_{t},...,X_{T})\in
A_{x}|X_{t}=x)=\mathbb{P}_{x}(A_{x}|B_{x})=\mathbb{P}_{x}(A_{x})\geq
1-\varepsilon,$ (3.1)

where the last equality follows from the fact that $B_{x}$ is independent of
$A_{x}$, because it only depends on the steps taken in the tree-coordinate and
hence it is independent of the steps in the $H$-coordinates between two tree-
coordinate steps. (To see this, note that the random walk path
$(Y_{1},...,Y_{R})$ by $\mathbb{P}_{x}$ could be generated by first generating
a suitable random walk path ${\cal T}$ in $\mathbb{T}^{d}$, and then adding a
suitably chosen random number of random $H$-steps in between every consecutive
pair of steps of ${\cal T}$, independently from each other and from ${\cal
T}$.) Since $x$ was arbitrary, from (3.1) we obtain

$\mathbb{P}((X_{t},...,X_{T})\in A_{X_{t}})\geq 1-\varepsilon.$ (3.2)

Let $U=D_{1},D_{2},...,D_{\ell}=D$ be the ray of bags between $U$ and $D$.
Denote by $t_{i}$ the first time that ${\cal X}$ enters $D_{i}$ after $\tau$
and let $r_{i}$ be the first time exiting $D_{i}$ after $t_{i}$. Finally, let
$A^{i}$ be the the event that $(X_{t_{i}},...,X_{T})$ visits every vertex of
$D_{i}$ before leaving $D_{i}$, in other words,
$\\{X_{t_{i}},...,X_{r_{i}}\\}=D_{i}$. Note that, conditional on
$\\{X_{t_{i}}\\}$ and $\\{X_{r_{i}}\\}$, the events $\\{A^{j}\\}$ are
independent, hence from the uniform lower bound (3.2) we have:

$\mathbb{P}\bigl{(}(\exists i\in[2,...,\ell-1]:\,\,A^{i}\mid
X_{t_{2}},...,X_{t_{l-1}},X_{r_{2}},...,X_{r_{l-1}}\bigr{)}\geq
1-\varepsilon^{\ell-2}.$

Using the law of total probability

$\mathbb{P}\bigl{(}(\exists i\in[2,...,\ell-1]:\,\,A^{i}\bigr{)}\geq
1-\varepsilon^{\ell-2}.$

We have just shown that $D$ is memorable for ${\cal X}$ with probability less
than $\varepsilon^{\ell-2}$. There are $d(d-1)^{s-1}$ vertices of $T_{n}$ with
distance $s$ from the root for all $1\leq s\leq n$, so

$\displaystyle\mathbb{P}({\cal X}\text{ has a memorable bag outside
}T_{m}\square wH)\leq\sum_{B\not\in T_{m}\square wH}\mathbb{P}(B\text{ is
memorable for }{\cal X})\leq$
$\displaystyle\sum_{s=m+1}^{n}\varepsilon^{s-2}d(d-1)^{s-1}=\frac{d}{\varepsilon}\sum_{s=m+1}^{n}(\varepsilon(d-1))^{s-1}<\frac{d}{\varepsilon}(\varepsilon(d-1))^{m}\frac{1}{1-\varepsilon(d-1)}.$

By definition $\varepsilon(d-1)<1$. The number on the right hand side does not
depend on $n$, thus we can choose $m$ big enough so that it is less than
$\alpha$. ∎

###### Proof of Theorem 3.1.

Let $W$ be as in Lemma 3.5.

Let $\alpha_{0}>0$ be arbitrary. We want to prove that if $w>W$ then for all
vertices $a,b$ of $G$, there exists an $m$ with

$\lim_{n\to\infty}\mathbb{P}(\text{LERW}_{T_{n}\square wH}(a\to b)\text{
leaves }T_{m}\square wH)\leq\alpha_{0}.$

This is equivalent with the definition of connectedness of the
$\mathsf{FSF}_{w}$ in ${\mathbb{T}^{d}\square H}$, because by Wilson’s
algorithm $\text{LERW}_{T_{n}\square wH}(a\to b)$ has the same distribution as
the path between $a$ and $b$ in $\mathsf{UST}(T_{n}\square wH)$. By Lemma
2.13., one may assume that $a$ and $b$ are in the same bag, $U$. Define $h$ as
the minimum of the probability over all pairs $x\not=y\in U$ that random walk
in $T_{n}\square wH$ started from $x$ hits $y$ before leaving $U$. Let $k$ be
a positive integer, chosen to satisfy $\mathbb{P}(\text{Geom}(h)\geq
k)<\alpha_{0}/2$, where $\text{Geom}(h)$ denotes a geometric random variable
of parameter $h$.

Choose $m$ as in Lemma 3.5, with $\alpha:=\alpha_{0}/2k$. Denote by ${\cal X}$
a random walk started from $a$ in $T_{n}\square wH$ and stopped when first
hitting $b$. One can construct ${\cal X}$ as follows. Start random walk from
$a$. If we hit $b$ before leaving $U$, we are finished, otherwise let
$X_{s_{1}}$ be the last step of this walk in $U$ before first leaving $U$.
Then consider the trip $(X_{s_{1}},...,X_{t_{1}})$. From the last vertex
$X_{t_{1}}\in U$ of this trip, continue the random walk until either hitting
$b$ or exiting $U$. The probability of the former is at least $h$; otherwise
let $X_{s_{2}}$ be the last vertex in $U$ before leaving $U$, and starting
from this vertex generate the trip $(X_{s_{2}},...,X_{t_{2}})$. Continue
similarly, until at some point we hit $b$ and at that point the construction
of ${\cal X}$ is finished. We see that after the end of every trip we had
probability at least $h$ to hit $b$, hence the total number of trips needed is
stochastically dominated by a geometric random variable of parameter $h$. Let
$J$ be such a random variable. If $\text{LE}({\cal X})$ intersects bag $D$,
then $D$ is memorable for one of the sub-trips of ${\cal X}$ by Proposition
3.4. The probability that a trip has a memorable bag outside of $T_{m}\square
wH$ is less than $\alpha$ by Lemma 3.5. A union bound gives us

$\mathbb{P}(\text{LE}({\cal X})\text{ leaves }T_{m}\square
cH)<\mathbb{P}(J>k)+k\alpha\leq\alpha_{0},$

completing the proof. ∎

###### Proof of Theorem 1.3.

The first part of the theorem is essentially Theorem 3.1.

For the second part, the case of small $w$, we have the same conditions on the
graph as in the unweighted case of Theorem 1.1 in [3] and one could repeat the
arguments therein, with minor modifications which we sketch next. The result
of Section 2 about random walk on the tree is obviously unchanged, while Lemma
3.1 also remains valid, with a different constant $b$, for the following
reason. Replace $2k^{6}$ in (3.2) by $2k^{6}/w$. Then the entire paragraph
containing (3.2) remains valid if we change every occurrence of $d$ to $wd$,
and that of $2k^{6}$ to $2k^{6}/w$. The rest of the proof of Lemma 3.1 in [3]
goes through without any change. The “second ingredient”, as explained after
the proof of Lemma 3.1, is based on the fact that random walk does not spend
much time in a bag. The key stochastic domination results are even “more true”
than in [3] when we have small weights on the $H$-edges, while the parts about
random walk within a bag, such as Lemma 3.2, remain unchanged. The rest of the
proof is automatically adapted to our setting. ∎

## Acknowledgement

The first three authors would like to thank the Rényi REU 2020 program for
undergraduate research. The third author is partially supported by the
ÚNKP-20-1 New National Excellence Program of the Ministry for Innovation and
Technology from the source of the National Research, Development and
Innovation Fund.
The last author was partially supported by the ERC Consolidator Grant 772466
“NOISE”, and by Icelandic Research Fund Grant 185233-051.

## References

  * [1] T. Hutchcroft and A. Nachmias. Indistinguishability of trees in uniform spanning forests. Probab. Theory Related Fields 168 (2017), 113–152.
  * [2] R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press, New York, 2016. Available at http://pages.iu.edu/~rdlyons/
  * [3] G. Pete and Á. Timár. The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set (2020), arXiv:2006.06387
  * [4] P. Tang. Weights of uniform spanning forests on nonunimodular transitive graphs (2019), arXiv:1908.09889
  * [5] Á. Timár. Indistinguishability of the components of random spanning forests. Ann. Probab. 46 (2018), 2221–2242.
  * [6] D. B. Wilson. Generating random spanning trees more quickly than the cover time. Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pp. 296–303., New York, 1996.

Marcell Alexy
Eötvös Lóránd University
H-1117 Budapest, Pázmány Péter sétány 1/C
alexy.marcell[at]hotmail.com
Márton Borbényi
Eötvös Lóránd University
H-1117 Budapest, Pázmány Péter sétány 1/C
marton.borbenyi[at]gmail.com
András Imolay
Eötvös Lóránd University
H-1117 Budapest, Pázmány Péter sétány 1/C
imolay.andras[at]gmail.com
Ádám Timár
Division of Mathematics, The Science Institute, University of Iceland
Dunhaga 3 IS-107 Reykjavik, Iceland
and
Alfréd Rényi Institute of Mathematics
Reáltanoda u. 13-15, Budapest 1053 Hungary
madaramit[at]gmail.com


# Estimation of the Parameters of Vector Autoregressive (VAR) Time Series
Model with Symmetric Stable Noise

Aastha M. Sathe and N. S. Upadhye
Email address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Centre of Excellence in Computational Mathematics and Data Science,
Department of Mathematics, Indian Institute of Technology Madras,
Chennai-600036, INDIA

(August 27, 2024)

###### Abstract

In this article, we propose the fractional lower order covariance method
(FLOC) for estimating the parameters of vector autoregressive process (VAR) of
order $p$, $p\geq 1$ with symmetric stable noise. Further, we show the
efficiency, accuracy and simplicity of our methods through Monte-Carlo
simulation.

Keywords: VAR model, stable distributions, parameter estimation, simulation

##  1 Introduction

The univariate stationary time series models, namely, the autoregressive
models (AR), moving average models (MA) and the general autoregressive moving
average models (ARMA) are popular tools in the statistical analysis of
univariate time series data (see [1]-[5]). On the other hand, for the analysis
of multivariate time series data, one of the most successful, flexible, and
easy to use models is the vector autoregressive (VAR) model. The VAR model is
especially useful for describing. In the classical definition, the above
mentioned models are assumed to be second-order due to finite second moment of
the noise term. However, these models fail to capture the heavy-tails of the
data. This motivates us to use the family of stable distributions to model the
data. Some of the significant and attractive features of stable distributions,
apart from stability are heavy-tails, leptokurtic shape, domains of
attraction, infinite second moment (with the exception of Gaussian case) and
skewness. For more details on stable distributions, see [12]. Hence, there is
a need to explore the behaviour of the above mentioned models with stable
noise for effective modelling of the time series data which also involves
estimation of the parameters of these models. However, because of the infinite
variance, only a handful of estimation techniques are available for models
based on stable noise (see [13]-[20]).

The structure of dependence for the stable-based models cannot be described by
the covariance or correlation functions (univariate case) and cross-covariance
or cross-correlation (multivariate case). However, one can find alternative
measures of dependence that can replace the classical ones in the case of
infinite variance. Some of them are: codifference, covariation and fractional
lower order covariance (FLOC) for one-dimensional models and cross-
codifference, cross-covariation and cross-FLOC for multidimensional models
(see [21]-[29]).

In this article, we develop a method for estimating the parameters of
multidimensional VAR model of order $p$, $p\geq 1$, with symmetric stable
noise for $\alpha\in[1.5,2]$. The proposed method employs the use of FLOC,
considered as an extension of the covariance function to the stable case and
with several interesting applications. The method is reasonable and effective
both from the theoretical and practical aspects. The efficiency of the method
is shown on the simulated data and by comparing it with the classical least
squares and Yule-Walker method for VAR models.

The paper is organized as follows. Section 2 gives a brief introduction to the
stable distributions and bidimensional VAR($p$) model along with the necessary
definitions and notations. Section 3 discusses the FLOC based parameter
estimation method. Section 4 deals with simulations and comparative analysis
of the proposed method with the classical Yule-Walker method. Section 5 deals
with an application to financial data. Finally, Section 6 gives some
concluding remarks on the proposed method.

##  2 Preliminaries and Notations

###  2.1 Stable Distributions

These distributions form a rich class of heavy-tailed distributions,
introduced by Paul Lévy [30], in his study on the Generalized Central Limit
Theorem. In the one-dimensional case, each distribution, in this class, is
characterized by four parameters, namely $\alpha$, $\beta$, $\sigma$, and
$\delta$, which, respectively, denote the index of stability, skewness, scale
and shift of the distribution. Their respective ranges are given by
$\alpha\in(0,2]$, $\beta\in[-1,1]$, $\sigma>0$ and $\delta\in\mathbb{R}$. For
more details, see [31]. The characteristic function representation for a
univariate stable random variable $Z$ is given by [12]

$\displaystyle\phi(\textit{t})$ $\displaystyle=$
$\displaystyle\begin{cases}\exp\left\\{-(\sigma|t|)^{\alpha}\left[1-\iota\beta{\rm
sign}(t)\tan\left(\frac{\pi\alpha}{2}\right)\right]+\iota\delta
t\right\\},&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \alpha\neq 1,\\\
\exp\left\\{-\sigma|t|\left[1+\iota\beta\frac{2}{\pi}{\rm
sign}(t)\ln|t|\right]+\iota\delta\textit{t}\right\\},&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\alpha=1,\end{cases}$ (1)

In this paper, we deal with symmetric stable distributions. We say that the
distribution is symmetric around zero if and only if $\beta=\delta=0$ in (1),
i.e., if the characteristic function is

$\phi(t)=\mathbb{E}(\exp itZ)=\exp\left\\{-(\sigma|t|)^{\alpha}\right\\}$ (2)

Note that when $\alpha=2$, $\beta=0$ and $\delta=0$, the distribution is
$S\alpha S$ Gaussian. Also, except the Gaussian case with $\alpha=2$, the
variance of $Z$ is infinite.

In the multi-dimensional case, the characteristic function of a symmetric
stable vector $\mathbf{Z}=(Z_{1},Z_{2},\cdots,Z_{r})$ takes the following form
[12]

$\mathbb{E}(\exp
i<\boldsymbol{t},\boldsymbol{Z}>)=\exp\Big{\\{}\int_{S_{r}}\mid<\boldsymbol{t},\boldsymbol{s}>\mid^{\alpha}\Gamma(ds)\Big{\\}}$
(3)

where $\Gamma(\cdot)$ is a finite spectral symmetric measure on the unit
sphere $S_{r}$ in $\mathbb{R}^{r}$ and $<\cdot,\cdot>$ is the inner product.
Thus, a necessary and sufficient condition for a stable vector to be symmetric
is that the shift vector $\delta_{\boldsymbol{0}}=\boldsymbol{0}$ and
$\Gamma(\cdot)$ is a finite spectral symmetric measure on $S_{r}$. Note that
the information about skewness and scale of the multi-dimensional stable
distributions are included in the spectral measure $\Gamma(\cdot)$.

###  2.2 Multidimensional VAR($p$) model with symmetric stable noise

Next, we discuss some important definitions and notations required for
parameter estimation of multidimensional vector autoregressive of order $p$
(VAR($p$)) model with symmetric stable noise. We begin our discussion with the
classical definitions of the second-order white noise and of the general
multidimensional VAR($p$) model which is later extended and modified to
incorporate the infinite-variance noise instead of the classical finite
variance white noise.

Let the multidimensional time series
$\\{\mathbf{Z}_{t}=(Z_{1t},\cdots,Z_{rt})^{T}:t\in\mathbb{Z}\\}$ be a white
noise process with mean $\boldsymbol{0}$ and covariance matrix $\sum$ if
$\\{\mathbf{Z}_{t}\\}$ is weak-sense stationary with mean vector
$\boldsymbol{0}$ and covariance matrix function given by [4]

$\displaystyle\gamma(h)=\mathbb{E}[\mathbf{Z}_{(t+h)}^{T}\mathbf{Z}_{t}]=\begin{cases}\sum\leavevmode\nobreak\
\leavevmode\nobreak\ \rm{when}\leavevmode\nobreak\ \textit{h}=0,\\\
0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \rm{otherwise}.\end{cases}$ (4)

Let the multidimensional time series
$\\{\mathbf{X}_{t}=(X_{1t},\cdots,X_{rt})^{T}:t\in\mathbb{Z}\\}$ (mean-
corrected) be a causal VAR($p$) process if it is weak-sense stationary and for
all $t\in\mathbb{Z}$ it satisfies the following equation ([4], [21])

$\mathbf{X}_{t}=A_{1}\mathbf{X}_{t-1}+\cdots+A_{p}\mathbf{X}_{t-p}+\mathbf{Z}_{t}$
(6)

where $\\{\mathbf{Z}_{t}\\}$ is a multidimensional white noise and
$A_{1}\cdots A_{p}$ are $r\times r$ matrices of the coefficients. Moreover,

$\det(I-A_{1}z-\cdots-A_{p}z^{p})\neq 0$ (7)

for all $z\in\mathbb{C}$ such that $|z|\leq 1$, where $I$ denotes an identity
matrix.
Equivalently, if there exists matrices $\\{\Psi_{j}\\}$ with absolutely
summable components such that for all $t\in\mathbb{Z}$

$\mathbf{X}_{t}=\sum_{j=0}^{\infty}\Psi_{j}\mathbf{Z}_{t-j}$ (8)

where the matrices $\\{\Psi_{j}\\}$ are found recursively

$\Psi_{j}=\Phi_{j}+\sum_{k=1}^{\infty}A_{k}\Psi_{j-k},\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ j=0,1,2,\cdots$ (9)

where $\Phi_{0}=I$, $\Phi_{j}=0$ for $j>0$, $A_{j}=0$ for $j>p$, $\Psi_{j}=0$
for $j<0$.

Let the multidimensional time series
$\\{\mathbf{X}_{t}=(X_{1t},\cdots,X_{rt})^{T}:t\in\mathbb{Z}\\}$ (mean-
corrected) be a causal VAR($p$) process with symmetric stable noise if for all
$t\in\mathbb{Z}$ it satisfies the following equation [21]

$\mathbf{X}_{t}=A_{1}\mathbf{X}_{t-1}+\cdots+A_{p}\mathbf{X}_{t-p}+\mathbf{Z}_{t}$
(10)

where the multidimensional noise $\\{\mathbf{Z}_{t}\\}$ is a symmetric stable
vector in $\mathbb{R}^{r}$ with the characteristic function defined in (3) and
$A_{1},\cdots,A_{p}$ are $r\times r$ matrices of the coefficients.
Additionally, we assume that $\mathbf{Z}_{t}$ is independent from
$\mathbf{Z}_{t+h}$ for all $h\neq 0$. Note that the causality of the model is
defined in the same way as in the classical VAR($p$) process defined above.

###  2.3 Measures of dependence for stable processes

Note that, due to undefined covariance when $\alpha<2$, the classical
dependence mesaures such as the autocovariance or autocorrelation function
cannot be considered as a tool for developing methods of parameter estimation
of the process defined in (10). In such case, alternative measures of
dependence are available in literature that can replace the classical
dependence measures. A few known choices are: normalized autocovariation,
autocodifference and fractional lower order covariance (FLOC). For details,
see ([21], Subsection 2.1 ). Amongst these choices, for the estimation of the
parameters of the process defined in (10), we prefer to use FLOC due to its
simple formulation and accuracy. Next, we present the definitions of FLOC and
the cross-FLOC estimator.

####  2.3.1 Fractional lower order covariance-FLOC

The fractional lower order covariance for a bidimensional symmetric stable
random vector $(X,Y)$ is defined as follows [32]

$\mathrm{FLOC}(X,Y,A,B)=\mathbb{E}[X^{<A>}Y^{<B>}]$ (11)

such that $A,B\geq 0$ satisfying $A+B<\alpha$.

Note that the term defined in (11) is dependent on the choice of $A$ and $B$
and in the Gaussian case, it reduces to the classical covariance when $A=B=1$.
The above measure is applicable to any symmetric stable vector, even with
$0<\alpha<1$.

Also, we observe that when $A=1,\leavevmode\nobreak\ B=q-1$, where
$q\in[1,\alpha)$ and $\alpha\in(1,2)$, the following relation holds between
FLOC and covariation function $\mathrm{CV}(X,Y)$ [21]

$\mathrm{FLOC}(X,Y,1,q-1)=\frac{\mathrm{CV}(X,Y)\mathbb{E}(|Y|^{q})}{q\sigma_{Y}^{\alpha}},$
(12)

where

$\mathrm{CV}(X,Y)=q\frac{\mathbb{E}(XY^{<q-1>})}{\mathbb{E}(|Y|^{q})}\sigma_{Y}^{\alpha},q\in[1,\alpha).$

######  Remark.

Both the measures of dependence for stable processes, namely,
$\mathrm{CV}(X,Y)$ and $\mathrm{FLOC}(X,Y,A,B)$ are not symmetric in its
arguments as opposed to the classical covariance function $\mathrm{Cov}(X,Y)$.

The FLOC is used as a measure of the spatio-temporal dependence of the
multidimensional process
$\\{\mathbf{X}_{t}=(X_{1t},\cdots,X_{rt})^{T}:t\in\mathbb{Z}\\}$. The
estimator of the cross-FLOC is defined as follows [21]

$\widehat{FLOC}(X_{it},X_{j(t-k)},A,B)=\frac{\sum_{n=L_{1}}^{L_{2}}|x_{in}|^{A}|x_{j(n-k)}|^{B}\mathrm{sign}(x_{in}x_{j(n-k)})}{L_{2}-L_{1}},\leavevmode\nobreak\
i,j=1,\cdots,r$ (13)

where $\\{x_{i1},x_{i2},\cdots,x_{iN}\\}$ and
$\\{x_{j1},x_{j2},\cdots,x_{jN}\\}$ are sample trajectories of length $N$
corresponding to the multivariate process $(X_{1t},\cdots,X_{rt})$ and
$L_{1}=\max(0,k)$, $L_{2}=\min(N,N+k)$. Note that in (13) replace $k$ with
$-k$ to obtain $\widehat{FLOC}(X_{it},X_{j(t+k)},A,B)$

##  3 FLOC based parameter estimation of multidimensional VAR($p$) process
with symmetric stable noise

In this section, we propose a new method for estimating the coefficients of
the $r\times r$ matrices $A_{1}\cdots A_{p}$ of a causal multidimensional
VAR($p$) process
$\\{\mathbf{X}_{t}=(X_{1t},\cdots,X_{rt})^{T}:t\in\mathbb{Z}\\}$ with
symmetric stable noise. The proposed method is based on FLOC introduced in the
previous section. FLOC is well defined for stable distributions and can be
used as a substitute of covariance function specially when the second moment
is infinite. The proposed method is discussed below.

  * •

Let $\\{X_{t}\\}$ be a multidimensional causal VAR($p$) process with symmetric
stable noise given by

$\mathbf{X}_{t}=A_{1}\mathbf{X}_{t-1}+\cdots+A_{p}\mathbf{X}_{t-p}+\mathbf{Z}_{t}$
(14)

where $\\{\mathbf{Z}_{t}\\}$ is a multidimensional symmetric stable noise with
$\alpha>1$.

  * •

Multiply both sides of (14) by
$\\{\mathbf{X}_{t-l}^{<B>^{T}}=(X_{1(t-l)}^{<B>},\cdots,X_{r(t-l)}^{<B>})\\}$
and taking expectation, we obtain the following expression

$\mathbb{E}[\mathbf{X}_{t}\mathbf{X}_{t-l}^{<B>^{T}}]=A_{1}\mathbb{E}[\mathbf{X}_{t-1}\mathbf{X}_{t-l}^{<B>^{T}}]+\cdots+A_{p}\mathbb{E}[\mathbf{X}_{t-p}\mathbf{X}_{t-l}^{<B>^{T}}]$
(15)

  * •

Next, define

$\mathbf{\Gamma}_{l}=FLOC(\mathbf{X}_{t},\mathbf{X}_{t-l},1,q-1)=\mathbb{E}[\mathbf{X}_{t}\mathbf{X}_{t-l}^{<q-1>^{T}}],\leavevmode\nobreak\
l=0,\pm 1,\pm 2,\cdots$ (16)

where $1\leq q<\alpha$ and $1<\alpha<2$. Thus, $\mathbf{\Gamma}_{l}$
represents $r\times r$ lag $l$ cross-FLOC matrix given as

$\mathbf{\Gamma}_{l}=\begin{bmatrix}\mathbb{E}[X_{1t}X_{1(t-l)}^{<q-1>}]&\dots&\mathbb{E}[X_{1t}X_{r(t-l)}^{<q-1>}]\\\
\vdots&\cdots&\vdots\\\
\mathbb{E}[X_{rt}X_{1(t-l)}^{<q-1>}]&\dots&\mathbb{E}[X_{rt}X_{r(t-l)}^{<q-1>}]\end{bmatrix}$

  * •

Thus, (15) takes the following form

$\mathbf{\Gamma}_{l}=A_{1}\mathbf{\Gamma}_{l-1}+\cdots+A_{p}\mathbf{\Gamma}_{l-p},\leavevmode\nobreak\
l=1,\cdots p$ (17)

and we get the following system of matrix equations

$[\mathbf{\Gamma}_{1}\cdots\mathbf{\Gamma}_{p}]=[A_{1}\cdots
A_{p}]\begin{bmatrix}\mathbf{\Gamma}_{0}&\mathbf{\Gamma}_{1}&\dots&\mathbf{\Gamma}_{p-1}\\\
\mathbf{\Gamma}_{-1}&\mathbf{\Gamma}_{0}&\dots&\mathbf{\Gamma}_{p-2}\\\
\vdots&\vdots&\cdots&\vdots\\\
\mathbf{\Gamma}_{1-p}&\mathbf{\Gamma}_{2-p}&\cdots&\mathbf{\Gamma}_{0}\end{bmatrix}$

  * •

Finally, the estimates of the coefficients of the $r\times r$ matrices
$A_{1}\cdots A_{p}$ are obtained from the expression given below using the
estimator of the cross-FLOC defined in (13).

$[\widehat{A}_{1}\cdots\widehat{A}_{p}]=[\mathbf{\widehat{\Gamma}}_{1}\cdots\mathbf{\widehat{\Gamma}}_{p}]\begin{bmatrix}\mathbf{\widehat{\Gamma}}_{0}&\mathbf{\widehat{\Gamma}}_{1}&\dots&\mathbf{\widehat{\Gamma}}_{p-1}\\\
\mathbf{\widehat{\Gamma}}_{-1}&\mathbf{\widehat{\Gamma}}_{0}&\dots&\mathbf{\widehat{\Gamma}}_{p-2}\\\
\vdots&\vdots&\cdots&\vdots\\\
\mathbf{\widehat{\Gamma}}_{1-p}&\mathbf{\widehat{\Gamma}}_{2-p}&\cdots&\mathbf{\widehat{\Gamma}}_{0}\end{bmatrix}^{-1}$

##  4 Simulation and Comparative Analysis

In this section, we investigate the performance of the proposed FLOC based
parameter estimation method for bidimensional VAR($p$) models with symmetric
stable noise through Monte Carlo simulations. Further, focussing on the
practical aspect, we compare the results of the proposed method with the
classical least squares (LS) and Yule-Walker (Y-W) [9] method to emphasize the
difference in the two approaches. Note that, $A=1$ is fixed throughout.

To test the proposed estimation procedure, we consider the bidimensional
VAR(2) model $\\{\mathbf{X}_{t}=(X_{1t},X_{2t})^{T}:t\in\mathbb{Z}\\}$ with
bidimensional independent symmetric stable noise
$\mathbf{Z}_{t}=(Z_{1t},Z_{2t})^{T}:t\in\mathbb{Z}\\}$ given below

$\mathbf{X}_{t}=A_{1}\mathbf{X}_{t-1}+A_{2}\mathbf{X}_{t-2}+\mathbf{Z}_{t}$
(18)

where $A_{1}=\begin{bmatrix}a_{1}&a_{3}\\\ a_{2}&a_{4}\end{bmatrix}$ and
$A_{2}=\begin{bmatrix}a_{5}&a_{7}\\\ a_{6}&a_{8}\end{bmatrix}$.
Next, for different values of the sample size $n$, $\alpha\in[1.5,2]$,
$A_{1}=\begin{bmatrix}0.1&0.3\\\ 0.2&0.1\end{bmatrix}$ and
$A_{2}=\begin{bmatrix}0.2&0.2\\\ 0.05&0.1\end{bmatrix}$, simulations are run
where 500 realisations of the model in (18) are generated using the function
varima.sim available in “portes” package in R. For each realisation, we
calculate $\widehat{A}_{1}$ and $\widehat{A}_{2}$ for different values of $B$.
From Tables 1, 2, 3 and 4, we observe that for all considered $n$, $\alpha$
and different values of $B$, the root mean squared errors (RMSE) get smaller
as $B$ approaches close to $\alpha-1$.

### Comparison of classical LS and Y-W with FLOC for VAR models

For comparison of our proposed method with the classical LS and Y-W method, we
use the VAR.est function available in the “VAR.etp” package and marfit
function available in the “TSSS” package in R respectively. Next, simulations
are run where 500 realisations of the model in (18), for different values of
$n$ and $\alpha$, are generated using the function varima.sim available in
“portes” package in R, where $A_{1}=\begin{bmatrix}0.1&0.3\\\
0.2&0.1\end{bmatrix}$ and $A_{2}=\begin{bmatrix}0.3&0.2\\\
0.4&0.1\end{bmatrix}$.
We then estimate the coefficients of the matrices $A_{1}$ and $A_{2}$ using
our proposed method and the classical LS and Y-W method. For the FLOC-based
estimators, we choose the parameter $B$ close to $\alpha-1$ as observed from
Tables 1, 2, 3 and 4, more specifically, $B=\alpha-1.05$. From Tables 5, 6 and
7, we observe that the RMSE of the estimates obtained via the classical LS and
Y-W method is fairly larger than our proposed FLOC method for different values
of $n$ and $\alpha$. Summarising, our proposed method works better than
classical LS and Y-W method.

B | 0.00 | 0.11 | 0.22 | 0.33 | 0.44 | 0.55
---|---|---|---|---|---|---
$a_{1}=0.1$ |  |  |  |  |  |
Mean | 0.1068 | 0.0973 | 0.1037 | 0.0983 | 0.1007 | 0.0984
RMSE | 0.1634 | 0.1244 | 0.1156 | 0.0923 | 0.0850 | 0.0856
$a_{2}=0.2$ |  |  |  |  |  |
Mean | 0.02057 | 0.1735 | 0.1983 | 0.1862 | 0.2016 | 0.2017
RMSE | 0.3383 | 0.3950 | 0.2358 | 0.2846 | 0.1297 | 0.1749
$a_{3}=0.3$ |  |  |  |  |  |
Mean | 0.3129 | 0.3046 | 0.3067 | 0.3047 | 0.3056 | 0.2925
RMSE | 0.1980 | 0.1887 | 0.1768 | 0.1754 | 0.1320 | 0.2206
$a_{4}=0.1$ |  |  |  |  |  |
Mean | 0.1288 | 0.1141 | 0.1173 | 0.1198 | 0.1216 | 0.982
RMSE | 0.1654 | 0.1343 | 0.1144 | 0.1642 | 0.0912 | 0.4188
$a_{5}=0.2$ |  |  |  |  |  |
Mean | 0.1750 | 0.1841 | 0.1834 | 0.1912 | 0.1945 | 0.1937
RMSE | 0.1529 | 0.1258 | 0.1141 | 0.0994 | 0.0866 | 0.0828
$a_{6}=0.05$ |  |  |  |  |  |
Mean | 0.0356 | 0.0558 | 0.0451 | 0.0443 | 0.0408 | 0.0442
RMSE | 0.2655 | 0.1969 | 0.1767 | 0.1966 | 0.1020 | 0.1524
$a_{7}=0.2$ |  |  |  |  |  |
Mean | 0.1947 | 0.2007 | 0.1959 | 0.2062 | 0.1913 | 0.1998
RMSE | 0.1848 | 0.1687 | 0.1503 | 0.1373 | 0.1156 | 0.1292
$a_{8}=0.1$ |  |  |  |  |  |
Mean | 0.0947 | 0.1010 | 0.0866 | 0.1053 | 0.0901 | 0.1062
RMSE | 0.2515 | 0.2027 | 0.1393 | 0.1292 | 0.0890 | 0.1679
Table 1: Mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ for $\alpha=1.6$ and $n=200$ B | 0.00 | 0.11 | 0.22 | 0.33 | 0.44 | 0.55
---|---|---|---|---|---|---
$a_{1}=0.1$ |  |  |  |  |  |
Mean | 0.0893 | 0.0983 | 0.0959 | 0.1006 | 0.1036 | 0.0989
RMSE | 0.1479 | 0.1000 | 0.0772 | 0.0578 | 0.0500 | 0.0393
$a_{2}=0.2$ |  |  |  |  |  |
Mean | 0.2410 | 0.1896 | 0.2022 | 0.1951 | 0.2034 | 0.2021
RMSE | 0.9466 | 0.4107 | 0.1085 | 0.1200 | 0.0882 | 0.0892
$a_{3}=0.3$ |  |  |  |  |  |
Mean | 0.3457 | 0.3097 | 0.3013 | 0.2970 | 0.3008 | 0.3020
RMSE | 0.6879 | 0.1852 | 0.1230 | 0.0791 | 0.1711 | 0.0768
$a_{4}=0.1$ |  |  |  |  |  |
Mean | 0.1374 | 0.1153 | 0.1199 | 0.1241 | 0.1203 | 0.1270
RMSE | 0.5535 | 0.0876 | 0.0670 | 0.0719 | 0.0880 | 0.0708
$a_{5}=0.2$ |  |  |  |  |  |
Mean | 0.1834 | 0.1950 | 0.1958 | 0.1981 | 0.1964 | 0.1981
RMSE | 0.2322 | 0.1022 | 0.0743 | 0.0596 | 0.0587 | 0.0429
$a_{6}=0.05$ |  |  |  |  |  |
Mean | 0.0164 | 0.0480 | 0.0416 | 0.0424 | 0.0404 | 0.0403
RMSE | 0.6018 | 0.2008 | 0.0808 | 0.0785 | 0.0662 | 0.0669
$a_{7}=0.2$ |  |  |  |  |  |
Mean | 0.1905 | 0.1948 | 0.2033 | 0.1970 | 0.2067 | 0.1974
RMSE | 0.3064 | 0.1265 | 0.1024 | 0.0720 | 0.0850 | 0.0575
$a_{8}=0.1$ |  |  |  |  |  |
Mean | 0.0660 | 0.0996 | 0.0972 | 0.1003 | 0.0977 | 0.0950
RMSE | 0.5399 | 0.1922 | 0.0738 | 0.0688 | 0.0600 | 0.0507
Table 2: Mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ for $\alpha=1.6$ and $n=700$ B | 0.12 | 0.24 | 0.36 | 0.48 | 0.60 | 0.72
---|---|---|---|---|---|---
$a_{1}=0.1$ |  |  |  |  |  |
Mean | 0.1023 | 0.0965 | 0.1019 | 0.0978 | 0.0988 | 0.0987
RMSE | 0.1066 | 0.0902 | 0.0853 | 0.0755 | 0.0701 | 0.0760
$a_{2}=0.2$ |  |  |  |  |  |
Mean | 0.2026 | 0.1873 | 0.1993 | 0.1930 | 0.2007 | 0.1996
RMSE | 0.1461 | 0.1499 | 0.1215 | 0.1464 | 0.0909 | 0.1037
$a_{3}=0.3$ |  |  |  |  |  |
Mean | 0.3040 | 0.3038 | 0.2968 | 0.3017 | 0.3026 | 0.2918
RMSE | 0.1281 | 0.1127 | 0.1153 | 0.1089 | 0.0959 | 0.1151
$a_{4}=0.1$ |  |  |  |  |  |
Mean | 0.1348 | 0.1287 | 0.1332 | 0.1352 | 0.1330 | 0.1285
RMSE | 0.1109 | 0.0996 | 0.0924 | 0.1072 | 0.0805 | 0.1611
$a_{5}=0.2$ |  |  |  |  |  |
Mean | 0.1855 | 0.1886 | 0.1892 | 0.1922 | 0.1965 | 0.1940
RMSE | 0.1016 | 0.0893 | 0.0848 | 0.0794 | 0.0717 | 0.0675
$a_{6}=0.05$ |  |  |  |  |  |
Mean | 0.0371 | 0.0473 | 0.0374 | 0.0380 | 0.0388 | 0.0407
RMSE | 0.01315 | 0.1044 | 0.1102 | 0.1152 | 0.0804 | 0.0951
$a_{7}=0.2$ |  |  |  |  |  |
Mean | 0.1951 | 0.2009 | 0.1952 | 0.2042 | 0.1930 | 0.2002
RMSE | 0.1223 | 0.1182 | 0.1069 | 0.1035 | 0.0917 | 0.0892
$a_{8}=0.1$ |  |  |  |  |  |
Mean | 0.0924 | 0.0937 | 0.0895 | 0.0993 | 0.0875 | 0.0985
RMSE | 0.1221 | 0.1090 | 0.0927 | 0.0849 | 0.0721 | 0.0836
Table 3: Mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ for $\alpha=1.75$ and $n=200$ B | 0.12 | 0.24 | 0.36 | 0.48 | 0.60 | 0.72
---|---|---|---|---|---|---
$a_{1}=0.1$ |  |  |  |  |  |
Mean | 0.0956 | 0.0979 | 0.0966 | 0.0987 | 0.1023 | 0.0989
RMSE | 0.0618 | 0.0578 | 0.0527 | 0.0438 | 0.0411 | 0.0342
$a_{2}=0.2$ |  |  |  |  |  |
Mean | 0.2031 | 0.1948 | 0.1996 | 0.1977 | 0.2014 | 0.2012
RMSE | 0.2535 | 0.1811 | 0.0661 | 0.0587 | 0.0547 | 0.0543
$a_{3}=0.3$ |  |  |  |  |  |
Mean | 0.3080 | 0.3032 | 0.2982 | 0.2979 | 0.3011 | 0.3026
RMSE | 0.1766 | 0.0810 | 0.0664 | 0.0529 | 0.0919 | 0.0512
$a_{4}=0.1$ |  |  |  |  |  |
Mean | 0.1319 | 0.1308 | 0.1336 | 0.1382 | 0.1344 | 0.1380
RMSE | 0.0729 | 0.0644 | 0.0575 | 0.0615 | 0.0707 | 0.0593
$a_{5}=0.2$ |  |  |  |  |  |
Mean | 0.1951 | 0.1973 | 0.1990 | 0.1984 | 0.1975 | 0.1974
RMSE | 0.0828 | 0.0585 | 0.0486 | 0.0442 | 0.0423 | 0.0362
$a_{6}=0.05$ |  |  |  |  |  |
Mean | 0.0405 | 0.0424 | 0.0404 | 0.0388 | 0.0387 | 0.0382
RMSE | 0.1281 | 0.1090 | 0.0557 | 0.0521 | 0.0502 | 0.0493
$a_{7}=0.2$ |  |  |  |  |  |
Mean | 0.2045 | 0.1979 | 0.2027 | 0.1970 | 0.2040 | 0.1972
RMSE | 0.0877 | 0.0679 | 0.0614 | 0.0500 | 0.0565 | 0.0449
$a_{8}=0.1$ |  |  |  |  |  |
Mean | 0.0898 | 0.0945 | 0.0951 | 0.0978 | 0.0956 | 0.0923
RMSE | 0.1254 | 0.0843 | 0.0488 | 0.0444 | 0.0432 | 0.0396
Table 4: Mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ for $\alpha=1.75$ and $n=700$ True Values | FLOC | LS | Y-W
---|---|---|---
$\boldsymbol{(n=100,\alpha=2,B=0.95)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.088 | 0.092 | 0.072 | 0.099 | 0.079 | 0.095
$a_{2}=0.2$ | 0.192 | 0.104 | 0.191 | 0.119 | 0.198 | 0.100
$a_{3}=0.3$ | 0.297 | 0.091 | 0.290 | 0.087 | 0.297 | 0.095
$a_{4}=0.1$ | 0.089 | 0.084 | 0.077 | 0.011 | 0.091 | 0.096
$a_{5}=0.3$ | 0.277 | 0.091 | 0.251 | 0.102 | 0.251 | 0.102
$a_{6}=0.4$ | 0.397 | 0.097 | 0.386 | 0.090 | 0.378 | 0.094
$a_{7}=0.2$ | 0.204 | 0.096 | 0.193 | 0.093 | 0.192 | 0.098
$a_{8}=0.1$ | 0.092 | 0.081 | 0.078 | 0.103 | 0.063 | 0.102
|  |  |  |  |  |
$\boldsymbol{(n=800,\alpha=2,B=0.95)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.096 | 0.032 | 0.096 | 0.033 | 0.095 | 0.033
$a_{2}=0.2$ | 0.198 | 0.036 | 0.203 | 0.033 | 0.200 | 0.035
$a_{3}=0.3$ | 0.300 | 0.032 | 0.296 | 0.029 | 0.300 | 0.032
$a_{4}=0.1$ | 0.098 | 0.028 | 0.094 | 0.030 | 0.096 | 0.032
$a_{5}=0.3$ | 0.298 | 0.032 | 0.299 | 0.032 | 0.296 | 0.033
$a_{6}=0.4$ | 0.400 | 0.032 | 0.401 | 0.025 | 0.399 | 0.032
$a_{7}=0.2$ | 0.201 | 0.033 | 0.202 | 0.029 | 0.200 | 0.034
$a_{8}=0.1$ | 0.098 | 0.027 | 0.095 | 0.034 | 0.095 | 0.033
Table 5: Comparison of mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ using FLOC, LS and Y-W method True Values | FLOC | LS | Y-W
---|---|---|---
$\boldsymbol{(n=200,\alpha=1.85,B=0.8)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.098 | 0.068 | 0.092 | 0.066 | 0.096 | 0.066
$a_{2}=0.2$ | 0.195 | 0.081 | 0.192 | 0.076 | 0.195 | 0.078
$a_{3}=0.3$ | 0.300 | 0.073 | 0.0296 | 0.070 | 0.299 | 0.076
$a_{4}=0.1$ | 0.087 | 0.069 | 0.098 | 0.060 | 0.100 | 0.060
$a_{5}=0.3$ | 0.286 | 0.069 | 0.279 | 0.070 | 0.272 | 0.074
$a_{6}=0.4$ | 0.403 | 0.073 | 0.396 | 0.067 | 0.391 | 0.070
$a_{7}=0.2$ | 0.196 | 0.073 | 0.194 | 0.069 | 0.190 | 0.071
$a_{8}=0.1$ | 0.098 | 0.059 | 0.085 | 0.066 | 0.081 | 0.068
|  |  |  |  |  |
$\boldsymbol{(n=700,\alpha=1.85,B=0.8)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.098 | 0.035 | 0.097 | 0.034 | 0.098 | 0.034
$a_{2}=0.2$ | 0.198 | 0.067 | 0.197 | 0.044 | 0.197 | 0.045
$a_{3}=0.3$ | 0.299 | 0.039 | 0.298 | 0.038 | 0.302 | 0.060
$a_{4}=0.1$ | 0.085 | 0.059 | 0.100 | 0.031 | 0.100 | 0.032
$a_{5}=0.3$ | 0.296 | 0.035 | 0.294 | 0.035 | 0.292 | 0.036
$a_{6}=0.4$ | 0.404 | 0.052 | 0.399 | 0.037 | 0.397 | 0.032
$a_{7}=0.2$ | 0.202 | 0.043 | 0.201 | 0.043 | 0.200 | 0.047
$a_{8}=0.1$ | 0.103 | 0.042 | 0.094 | 0.035 | 0.093 | 0.036
Table 6: Comparison of mean and RMSE of 500 estimates of coefficients of $A_{1}$ and $A_{2}$ using FLOC, LS and Y-W method True Values | FLOC | LS | Y-W
---|---|---|---
$\boldsymbol{(n=300,\alpha=1.65,B=0.6)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.098 | 0.059 | 0.095 | 0.081 | 0.095 | 0.051
$a_{2}=0.2$ | 0.200 | 0.109 | 0.199 | 0.073 | 0.199 | 0.071
$a_{3}=0.3$ | 0.298 | 0.073 | 0.295 | 0.078 | 0.295 | 0.079
$a_{4}=0.1$ | 0.056 | 0.118 | 0.093 | 0.053 | 0.094 | 0.038
$a_{5}=0.3$ | 0.293 | 0.062 | 0.292 | 0.054 | 0.287 | 0.059
$a_{6}=0.4$ | 0.410 | 0.093 | 0.395 | 0.073 | 0.389 | 0.093
$a_{7}=0.2$ | 0.195 | 0.122 | 0.190 | 0.181 | 0.183 | 0.0184
$a_{8}=0.1$ | 0.116 | 0.077 | 0.092 | 0.052 | 0.089 | 0.055
|  |  |  |  |  |
$\boldsymbol{(n=600,\alpha=1.65,B=0.6)}$ | Mean | RMSE | Mean | RMSE | Mean | RMSE
$a_{1}=0.1$ | 0.101 | 0.041 | 0.097 | 0.034 | 0.099 | 0.035
$a_{2}=0.2$ | 0.201 | 0.083 | 0.200 | 0.094 | 0.198 | 0.058
$a_{3}=0.3$ | 0.301 | 0.047 | 0.301 | 0.041 | 0.300 | 0.045
$a_{4}=0.1$ | 0.053 | 0.107 | 0.100 | 0.033 | 0.100 | 0.034
$a_{5}=0.3$ | 0.297 | 0.040 | 0.296 | 0.032 | 0.294 | 0.037
$a_{6}=0.4$ | 0.412 | 0.064 | 0.397 | 0.065 | 0.396 | 0.043
$a_{7}=0.2$ | 0.197 | 0.048 | 0.195 | 0.040 | 0.196 | 0.037
$a_{8}=0.1$ | 0.116 | 0.066 | 0.095 | 0.043 | 0.095 | 0.036
Table 7: Comparison of mean and RMSE of 500 estimates of coefficients of
$A_{1}$ and $A_{2}$ using FLOC, LS and Y-W method

##  5 Application to Financial Data

In this section, we give an empirical analysis of the real financial bivariate
data using the vector autoregressive model with symmetric stable noise. We
consider the monthly simple returns of the stocks of International Business
Machines (IBM) and the SP Composite index (SP) from January 1961 to December
2011 with 612 observations. The datasets are available in the “MTS” package in
R.

To determine whether the considered time series datasets are stationary, we
implement the Augmented Dickey-Fuller Test (ADF) “adf.test” available in the
“tseries” package in R. The $p$-values obtained are 0.01 which confirms the
stationarity of the datasets. Next, we fit our proposed FLOC based
bidimensional autoregressive model with $p=2$ to the data. Since $A=1$ is
fixed, we need to estimate $\alpha$ from the bivariate data to find the value
of $B$. We use the hybrid method [31] to obtain the estimate of $\alpha$.
Thus, $\hat{\alpha}=1.86$ and $B=0.8$, value close to $1-\hat{\alpha}$ as
observed in the simulation study. The estimates of the coefficient matrices
are as follows: $\widehat{A}_{1}=\begin{bmatrix}0.003&0.069\\\
0.014&0.023\end{bmatrix}$ and $\widehat{A}_{2}=\begin{bmatrix}-0.040&0.031\\\
0.021&0.020\end{bmatrix}$.
Next, in order to check if the fitted model is appropriate we analyse the
residuals of the time series $\\{X_{1t}\\}$ and $\\{X_{2t}\\}$. In our model,
we assume that the noise series $\\{Z_{1t}\\}$ or $\\{Z_{2t}\\}$ is a sample
of independent and stable distributed random variables. Thus, we fit stable
distribition using the hybrid method [31] to the residual time series
$\\{Z_{1t}\\}$ and $\\{Z_{2t}\\}$. The estimates obtained are
($\hat{\alpha},\hat{\beta},\hat{\sigma},\hat{\delta})=(1.900,-0.708,0.027,0.008)$
and $1.851,0.148,0.035,0.010$), respectively for $\\{Z_{1t}\\}$ and
$\\{Z_{2t}\\}$.

In order to check if the residuals can be considered as an independent sample,
we make use of the empirical auto-FLOC function defined in [34] instead of the
classical autocovariance (autocorrelation) function for our model. From the
residuals auto-FLOC plot in Figure 1 and 2, one can observe that the
dependence in the residual series is almost unidentifiable. Thus, we assume
that $\\{Z_{1t}\\}$ and $\\{Z_{2t}\\}$ are independent for all values of $t$.
Note that for empirical auto-FLOC function for $\\{Z_{1t}\\}$, $A=1$ and
$B=0.85$ while for $\\{Z_{2t}\\}$, $A=1$ and $B=0.8$. Additionally, the QQ
plots as shown in Figure 3, represent the quantiles for fitted stable
distribution with the estimated parameters from the residuals and the
empirical quantiles for residuals. We observe the distribution of both the
residuals is almost stable. Finally, we perform the Kolmogorov-Smirnov (KS)
test as discussed in [33] with the hypothesis that $\\{Z_{1t}\\}$ and
$\\{Z_{2t}\\}$ have stable distribution. Since the obtained $p$-values are
0.60 and 0.7, respectively for $\\{Z_{1t}\\}$ and $\\{Z_{2t}\\}$, calculated
on the basis of 100 Monte Carlo repetitions, we fail to reject the hypothesis
at the significance level 0.05.

Figure 1: Residuals and Auto-FLOC Plot of $Z_{1t}$

Figure 2: Residuals and Auto-FLOC Plot of $Z_{2t}$

Figure 3: QQPlots of $Z_{1t}$ and $Z_{2t}$

##  6 Conclusion

To conclude, we make the following observations in relation to our proposed
methods.

  * •

In this article, a new estimation method for multidimensional VAR($p$), $p\geq
1$ with symmetric stable distribution ($\alpha\in[1.5,2])$ is introduced.

  * •

The proposed method is an extension of the classical Y-W method which is based
on the covariance function of the underlying process.

  * •

The simulation study reflects the effectiveness of the proposed method in
comparison to the classical LS and Y-W method. The application of the FLOC
measure is justified from the theoretical point of view in the considered case
(the theoretical covariance does not exist) however by simulation study we
have proved it is reasonable to use the new technique taking under account the
practical aspects.

  * •

Finally, we fit our proposed model to the bivariate financial data.

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