The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts

TL;DR

This paper derives the precise asymptotic behavior of the tail distribution for the collision time of multiple independent Brownian particles with different drifts, providing explicit formulas for the asymptotic parameters.

Contribution

It establishes the exact asymptotics of the collision time tail distribution for Brownian particles with different drifts, including explicit expressions for the constants involved.

Findings

Asymptotic formula for P_x(τ > t) as t→∞

Explicit expressions for constants C, h(x), α, γ

Identification of parameters in terms of drifts

Abstract

In this note we consider the time of the collision $\tau$ for $n$ independent Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact asymptotics of $P_x(\tau>t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and identify $C,h(x),\alpha,\gamma$ in terms of the drifts.

Equations

$$\tau =\inf \{t>0:{\mathbf{X}}_t\notin W\}.$$

$$\mathrm{IP}(\tau >t;{\mathbf{X}}_t\in \mathrm{d}\mathbf{y})=\det \left[p_t(x_i,y_j)\right]\mathrm{d}\mathbf{y},$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)={\int }_W\det \left[p_t(x_i,y_j)\right]\mathrm{d}\mathbf{y}.$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)\sim D\Delta (\mathbf{x})t^{-n(n-1)/4},$$

$$D={\left(2\pi \right)}^{-n/2}{c}_n{\int }_W{e}^{-\frac{{|\mathbf{y}|}^2}{2}}\Delta \left(\mathbf{y}\right)d\mathbf{y},$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)={\int }_{t/2}^{\infty }\frac{x}{\sqrt{2\pi }{s}^{3/2}}\exp \left[-\frac{(x+as{)}^2}{2s}\right]\mathrm{d}s,$$

$$\displaystyle{\rm IP}_{\boldsymbol{x}}(\tau>t)=\left\{\begin{array}[]{ll}\frac{2^{5/2}}{a^{2}\sqrt{2\pi}}xe^{ax}t^{-3/2}e^{-ta^{2}/4}\,\,(1+o(1))\;,&a_{1}>a_{2}\\[8.5359pt] \frac{2^{\frac{3}{2}}}{\sqrt{2\pi}}xt^{-\frac{1}{2}}\,\,(1+o(1))\;,&a_{1}=a_{2}\\[8.5359pt] 1-e^{-ax}+o(1)&a_{1}<a_{2}\;.\end{array}\right.$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)=$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)$$

$$(a_1,\dots ,a_{{\nu }_1})(a_{{\nu }_1+1},\dots ,a_{{\nu }_1+{\nu }_2}),\dots ,(a_{{\nu }_1+\dots +{\nu }_{q-1}+1},\dots ,a_{{\nu }_1+\dots +{\nu }_q}).$$

$$\left.\begin{array}[]{ccc}\bar{a}_{[1;1]}&>&\bar{a}_{[2;n]}\\ \bar{a}_{[1;2]}&>&\bar{a}_{[3;n]}\\ \vdots&\vdots&\vdots\\ \bar{a}_{[1;n-1]}&>&\bar{a}_{[n;n]}\end{array}\right\}\;.$$

$$f_i=f^k,\mathrm{if}{m}_{k-1}<i\le m_k.$$

$$f^1\le f^2\le \dots \le f^q$$

$$m_i^{\prime }=\inf \{m_j>m_{i-1}^{\prime }:m_j\in \mathbf{m},f_{m_j}<f_{m_{j+1}}\}.$$

$$i_k=\inf \{j>i_{k-1}:f_{m_j}<f_{m_{j+1}}\}.$$

$$f_{m_1^{\prime }}<f_{m_2^{\prime }}<\cdots <f_{m_{q^{\prime }}^{\prime }}.$$

$$\left.\begin{array}[]{ccccc}\bar{a}_{[1;1]}&>&f_{n}&>&\bar{a}_{[2;n]}\\ \bar{a}_{[1;2]}&>&f_{n}&>&\bar{a}_{[3;n]}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \bar{a}_{[1;n-1]}&>&f_{n}&>&\bar{a}_{[n;n]}\end{array}\right\}\;.$$

$${\bar{a}}_{(m_{i-1};m_{i-1}+k]}\ge f_{m_i}.$$

$${\bar{a}}_{[m_{i-1}^2+1;k]}^2>{\bar{a}}_{[k+1;m_i^2]}^2$$

$$(f^1{)}^i={\bar{a}}_{[m_{i-1}^2+1;m_i^1]}>{\bar{a}}_{[m_i^1+1;m_i^2]}\ge (f^1{)}^{i+1},$$

$$(f^1{)}^i={\bar{a}}_{[m_{i-1}^1+1;m_i^1]}>{\bar{a}}_{[m_i^1+1;m_i^2]}\ge (f^1{)}^{i+1},$$

$$\gamma =\frac{1}{2}\sum _{\ell =1}^q\left(\frac{1}{{\nu }_{\ell }}\sum _{m_{l-1}<u<v\le m_l}(a_u-a_v{)}^2\right),$$

$$\alpha =\frac{1}{2}\left(\sum _{j=1}^{q^{\prime }}\left(\genfrac{}{}{0ex}{}{{\nu }_j^{\prime }}{2}\right)+(n-q)+\sum _{j=1}^q\left(\genfrac{}{}{0ex}{}{{\nu }_j}{2}\right)\right),$$

$$h(\mathbf{x})$$

$$I(\mathbf{a},t)$$

$$\frac{1}{{\nu }_l}\sum _{m_{l-1}<u<v\le m_l}(a_u-a_v{)}^2=\sum _{m_{l-1}<u<v\le m_l}(a_u-{\bar{a}}^l{)}^2,$$

$${\bar{a}}^l=\frac{1}{{\nu }_l}\sum _{u=m_{l-1}+1}^{m_l}{a}_u.$$

$${\mathrm{IP}}_{\mathbf{x}}(\tau >t)$$

$$H(s_1,\dots ,s_{\ell })=\prod _{1\le i\le j\le \ell +1}(s_i+\dots +s_{j-1}).$$

Full text

Keywords: Brownian motion with drift, collision time.

AMS 2000 Subject Classification: Primary: 60J65.

11footnotetext: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland22footnotetext: This work was partially supported by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme: Programme HANAP under contract number MTKD-CT-2004-13389. 33footnotetext: This work was partially supported by This work was partially supported by KBN Grant N201 049 31/3997 (2007).

1 Introduction and results

Let $W=\{\boldsymbol{y}:y_{1}<\ldots<y_{n}\}$ be the Weyl chamber. Consider $\boldsymbol{X}_{t}=(X^{1}_{t},\ldots,X^{n}_{t})$, wherein coordinates are independent Brownian motions with unit variance parameter, drift vector $\boldsymbol{a}=(a_{1},\ldots,a_{n})$ and starting point $\boldsymbol{X}_{0}=\boldsymbol{x}\in W$. In this paper we study the collision time $\tau$, which is the exit time of $\boldsymbol{X}_{t}$ from the Weyl chamber, i.e.

[TABLE]

For identical drifts $a_{1}=\ldots=a_{n}$ , say $a_{i}\equiv 0$, the celebrated Karlin-McGregor formula states (see [7])

[TABLE]

where $p_{t}(x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^{2}}{2t}}$, which yields the tail distribution of $\tau$:

[TABLE]

For the use of Karlin-McGregor formula it is essential that processes $X_{t}^{1},\ldots,X^{n}_{t}$ are independent copies of the same strong Markov, with skip-free realizations process, starting at $t=0$ from $\boldsymbol{x}\in W$. In this case the asymptotic of ${\rm IP}_{\boldsymbol{x}}(\tau>t)$ was first studied by Grabiner [5] (for the Brownian case) (see also proofs by Doumerc and O’Connell [4] and Puchała [9]) Later Puchała & Rolski [10]) showed that this asymptotic is also true for the Poisson and continuous time random walk case. The above mentioned asymptotics is:

[TABLE]

where $\Delta(\boldsymbol{x})=\det\left[\left(x_{i}^{(j-1)}\right)_{i,j=1}^{n}\right]$ is the Vandermonde determinant, and

[TABLE]

for $t\to\infty$. Here and below $1/c_{n}=\prod_{j=1}^{n-1}j!$.

In this note we study the same problem, however for Brownian motions with different drifts. For this we derive first, in Section 2, a formula for ${\rm IP}_{\boldsymbol{x}}(\tau>t)$ by the change of measure. It is apparent that possible results must depend on the form of drift vector $\boldsymbol{a}$. For example we can analyze all cases for $n=2$, because in this case the collision equals to the first passage to zero of the Brownian process $X^{2}_{t}-X^{1}_{t}$, for which the density function is known (see e.g. [3]). Hence

[TABLE]

where $x=x_{2}-x_{1}$ and $a=a_{2}-a_{1}$. This yields

[TABLE]

For general $n$ the situation is much more complex and different scenarios are possibles. For example the drifts can be diverging and then ${\rm IP}_{\boldsymbol{x}}(\tau>t)$ tends to a positive constant, which the situation was analyzed by Biane et al [2]. Another case is when all drifts are equal, in which the case the probability ${\rm IP}_{\boldsymbol{x}}(\tau>t)$ is polynomially decaying, as it was found by Grabiner [5]. However there are various situations when the probabilities are exponentially decaying with polynomial prefactors. The full characterization depends on a concept of the stable partition of the drift vector, which the notion is introduced in Section 3. In Section 4 we state the main theorem, which shows all possible exact asymptotics of ${\rm IP}_{\boldsymbol{x}}(\tau>t)$ in from of $Ch(\boldsymbol{x})t^{-\alpha}e^{-\gamma t}$, where formulas for $C$,$\alpha$ and $\gamma$ are given in terms of the stable partition of the drift vector.

2 Formula for ${\rm IP}_{\boldsymbol{x}}(\tau>t)$.

We note our basic probabilistic space with natural history filtration $(\Omega,{\cal F},({\cal F}_{t}),{\rm IP}_{\boldsymbol{x}})$ and consider on it process $\boldsymbol{X}_{t}$ as defined in the Introduction. Unless otherwise stated we tacitly assume that $\boldsymbol{x}\in W$. We start off a lemma on the change of measure for the Brownian case, which the proof can be found for example in Asmussen [1], Theorem 3.4. Let $M_{t}=e^{<\boldsymbol{\alpha},\boldsymbol{X}_{t}>}/{\rm IE}e^{<\boldsymbol{\alpha},\boldsymbol{X}_{t}>}$ be a Wald martingale. For a probability measure ${\rm IP}_{\boldsymbol{x}}$ its restriction to ${\cal F}_{t}$ we denote by ${\rm IP}_{\boldsymbol{x}|t}$. Let $\tilde{{\rm IP}}_{\boldsymbol{x}}$ be a probability measure obtained by the change of measure ${\rm IP}_{\boldsymbol{x}}$ with the use of martingale $M_{t}$, that is defined by a family of measures $\tilde{{\rm IP}}_{\boldsymbol{x}|t}=M_{t}\,{\,{\rm d}}{\rm IP}_{\boldsymbol{x}|t}$, $t\geq 0$. For the theory we refer e.g. to Section XIII.3 in [1]

Lemma 2.1

If $\boldsymbol{X}_{t}$ is a Brownian motion with drift $\boldsymbol{a}$ under ${\rm IP}_{\boldsymbol{x}}$, then this process is a Brownian motion with drift $\boldsymbol{a}+\boldsymbol{\alpha}$ under $\tilde{{\rm IP}}_{\boldsymbol{x}}$.

The sought for formula for the tail distribution of the collision time is given in the next proposition.

Proposition 2.2

[TABLE]

*Proof. *We use $\boldsymbol{\alpha}=-\boldsymbol{a}$ to eliminate the drift under $\tilde{\rm IP}_{\boldsymbol{x}}$. Thus ${\rm IP}_{\boldsymbol{x}}(\tau>t)=\tilde{{\rm IE}}_{\boldsymbol{x}}[M^{-1}_{t};\tau>t]\;$. Now by Karlin-McGregor formula (1.1) we write

[TABLE]

and next, algebraic manipulations yield (2.5).

$\Box$

In the paper we use the following vector notations. For a vector $\boldsymbol{a}\in{\rm IR}^{n}$ we denote $\boldsymbol{a}_{[i,j]}=(a_{i},a_{i+1},\ldots,a_{j})$ and $\bar{a}_{[i,j]}=(a_{i}+a_{i+1}+\ldots+a_{j})/(j-i+1)$. We also use $\boldsymbol{a}_{(i,j]}=(a_{i+1},\ldots,a_{j})$ and $\boldsymbol{a}_{(i,j)}=(a_{i+1},\ldots,a_{j-1})$. By $\boldsymbol{z}^{\boldsymbol{k}}$, where $\boldsymbol{z}=(z_{1},\ldots,z_{m})$ and $\boldsymbol{k}=(k_{1},\ldots,k_{m})$ we denote $\prod_{j=1}^{m}z_{j}^{k_{j}}$.

3 Stable partition of $\boldsymbol{a}$.

Let $\boldsymbol{a}\in{\rm IR}^{n}$. Our aim is to make a suitable partition

[TABLE]

of $\boldsymbol{a}$, where $\nu_{i}>0$. For short we denote $m_{1}=\nu_{1},m_{2}=\nu_{1}+\nu_{2},\ldots,m_{q}=\nu_{1}+\ldots+\nu_{q}=n$. We also set $m_{0}=0$.

We say that sequence $\boldsymbol{a}$ is irreducible if

[TABLE]

Suppose we have a partition defined by $m_{1},\ldots,m_{q}$ . The mean of the $i^{\text{th}}$ sub-vector is denoted by $f^{i}=\bar{a}_{(m_{i-1};m_{i}]}$. Furthermore we define a vector $\boldsymbol{f}=(f_{1},\dots,f_{n})$ by

[TABLE]

It is said that partition (3.6) of vector $\boldsymbol{a}$ is stable if

[TABLE]

and each vector $\boldsymbol{a}_{(m_{i-1},m_{i}]}$ is irreducible $(i=1,\ldots,q)$. Remark that a stable partition is defined if we know $\boldsymbol{m}=(m_{1},\ldots,m_{q})$ for which (3.8) hold and each $\boldsymbol{a}_{(m_{i-1},m_{i}]}$ is irreducible $(i=1,\ldots,q)$. In the sequel, for a given stable partition of $\boldsymbol{a}$, characters $q,\boldsymbol{f},\boldsymbol{m}$ are reserved for it.

Consider now $f_{m_{1}},f_{m_{2}},\ldots,f_{m_{q}}$ and define a subsequence $\boldsymbol{m}^{{}^{\prime}}=(m^{\prime}_{1},\ldots,m{{}^{\prime}}_{q^{\prime}})$ of $\boldsymbol{m}=(m_{1},m_{2},\dots,m_{q})$ as follows. Let $q^{\prime}$ be the number of strict inequalities in $f^{1}\leq f^{2}\leq\ldots\leq f^{q}$ plus 1. Furthermore we define inductively by $m^{\prime}_{0}=0$ and for $i=1,\ldots,q^{\prime}-1$

[TABLE]

and finally we set $m_{q^{\prime}}=n$. We also define a subsequence of indices $i_{0},i_{1},\dots,i_{q^{\prime}}$ inductively by $i_{0}=0$ and

[TABLE]

Hence we have

[TABLE]

In this case we say that $(m^{\prime}_{1},\dots,m^{\prime}_{q^{\prime}})$ is a strong representation of the stable partition of $\boldsymbol{a}$ and $q^{\prime},\ (m^{\prime}_{1},\dots,m^{\prime}_{q^{\prime}})$ are characters reserved for it. Set $\nu^{\prime}_{i}=m^{\prime}_{i}-m^{\prime}_{i-1}$, $(i=1,\ldots,q^{\prime})$.

Example 1 Suppose that $\boldsymbol{a}=(3,1,2,5,1)$. Then $q=3$ and $m_{1}=2,m_{2}=3,m_{3}=5$ define the stable partition $(3,1)(2)(5,1)$ with means $f^{1}=2,f^{2}=2,f^{3}=3$. furthermore $q^{\prime}=2$, $m_{1}^{\prime}=3,m_{2}^{\prime}=5$ and $i_{1}=2,i_{2}=5$.

Proposition 3.1

For each vector $\boldsymbol{a}$, there exists its unique stable partition.

Before we state a proof of Proposition 3.1 we prove few lemmas.

Lemma 3.2

If $\boldsymbol{a}=(a_{1},\ldots,a_{n})$ is irreducible, then

[TABLE]

Proof. $f_{n}$ is a nontrivial weighted mean of every pair $\bar{a}_{[1;i]}$ and $\bar{a}_{[i+1;n]}$.

$\Box$

Lemma 3.3

In a stable partition, for each element $\boldsymbol{a}_{(m_{i-1};m_{i}]}$

[TABLE]

*Proof. *The case $k\leq m_{i}-m_{i-1}$ follows from Lemma 3.2. Clearly for $k=m_{i}-m_{i-1}$ we have equality. Consider now $k>m_{i}-m_{i-1}$. Than $\bar{a}_{(m_{i-1};m_{i-1}+k]}$ is a weighted mean of $f_{m_{i}}$ and $\bar{a}_{(m_{i};m_{i}+k-(m_{i}-m_{i-1})]}$ and the later term is greater or equal than $f_{m_{i}}$ by (3.8) and (3.9).

$\Box$

In the next lemma we consider two vectors $\boldsymbol{a}_{1}\in{\rm IR}^{n_{1}}$ and $\boldsymbol{a}_{2}\in{\rm IR}^{n_{2}}$. The corresponding $f$-s are $f_{n_{1}}$ and $f_{n_{2}}$ respectively. We consider a situation of creating a new vector $(\boldsymbol{a}_{1},\boldsymbol{a}_{2})=(a_{1}\ldots,a_{n_{1}+n_{2}})\in{\rm IR}^{n_{1}+n_{2}}$.

Lemma 3.4

Suppose that $\boldsymbol{a}_{1}$ and $\boldsymbol{a}_{2}$ are irreducible and $f_{n_{1}}>f_{n_{2}}$. Then vector $(\boldsymbol{a}_{1},\boldsymbol{a}_{2})$ is irreducible.

*Proof. *Recall that $(\boldsymbol{a}_{1},\boldsymbol{a}_{2})=(a_{1}\ldots,a_{n_{1}+n_{2}})\in{\rm IR}^{n_{1}+n_{2}}$. Suppose $1\leq k\leq n_{1}$. By Lemma 3.2 we have $\bar{a}_{[1;k-1]}>f_{n_{1}}>\bar{a}_{[k;n_{1}]}$, also $\bar{a}_{[1;k-1]}>f_{n_{1}}>f_{n_{2}}$. Hence $\bar{a}_{[1;k-1]}>\bar{a}_{[k;n_{1}+n_{2}]}$ becasue $\bar{a}_{[k;n_{1}+n_{2}]}$ is a weighted mean of $\bar{a}_{[k;n_{1}]}$ and $f_{n_{2}}$. Suppose now $n_{1}<k$. Then $\bar{a}_{[1;k-1]}$ is a weighted mean of $f_{n_{1}}$ and $\bar{a}_{[n_{1}+1,k]}$ and both by Lemma 3.2 are greater than $\bar{a}_{[k;n_{1}+n_{2}]}$, which completes the proof.

$\Box$

Proof of Proposition 3.1. The existence part is by induction with respect $n$. For $n=2$ we have two situations

if $a_{1}\leq a_{2}$, than $q=2$ with $m_{1}=1,\,\,m_{2}=2$ is a stable partition, 2. 2.

if $a_{1}>a_{2}$, than $q=1$ with $m_{1}=2$ is a stable partition.

Assume that there exists a stable partition with $q$ partition vectors of a vector $\boldsymbol{a}\in R^{n}$. We add a new element $a_{n+1}$ at the end of vector $\boldsymbol{a}$ to create new one $(\boldsymbol{a},a_{n+1})=(a_{1},\ldots,a_{n+1})$.

We have two situations.

If $a_{n+1}\geq f^{q}$ than in a stable partition $a_{n+1}$ is alone in the $q+1$ partition vector. 2. 2.

If $a_{n+1}<f^{q}$, than we proceed inductively as follow. We use Lemma 3.4 with $\boldsymbol{a}_{1}=\boldsymbol{a}_{[m_{q-1};m_{q}]}$ and $\boldsymbol{a}_{2}=(a_{n+1})$ and let $f^{q}$ and $f^{q+1}=a_{n+1}$ are means of these partition vectors. In result $(\boldsymbol{a}_{(m_{q-1};m_{q}]},a_{n+1})$ form an irreducible vector, for which we have to check whether condition (3.8) holds. If yes, then we end with a stable partition, otherwise we join the $q-1$ partition vector with the new $q$ partition vectors and repeat the procedure. In the worst case we end up with one partition vector.

For the uniqueness proof , suppose that we have two different stable partitions: $m_{1}^{1}<m_{2}^{1}<\dots<m_{q_{1}}^{1}$ and $m_{1}^{2}<m_{2}^{2}<\dots<m_{q_{2}}^{2}$. The means of $f$s are $(f^{1})^{1},\dots,(f^{1})^{q_{1}}$

for the first partition vector and $(f^{2})^{1},\dots,(f^{2})^{q_{2}}$ for the second respectively. Since partitions are supposely different, there exists $i$ such that $m_{i}^{1}\neq m_{i}^{2}$. We take the minimal $i$ with this property and without loss of generality we can assume $m_{i}^{2}>m_{i}^{1}$. Set $k=m_{i}^{2}-m_{i}^{1}$. We have to analaze the following cases.

$(m_{i}^{2}=m_{i+1}^{1})$. We have

[TABLE]

On the other hand $(f^{1})^{i}=\bar{a}_{[m^{1}_{i-1}+1;m_{i}^{1}]}=\bar{a}_{[m^{2}_{i-1}+1;m_{i}^{1}]}>\bar{a}_{[m^{1}_{i}+1;m_{i}^{2}]}=\bar{a}_{[m^{1}_{i}+1;m_{i+1}^{1}]}=(f^{2})^{i}$ and this contradics with $(f^{1})^{i}\leq(f^{1})^{i+1}$. 2. 2.

$(m_{i}^{2}>m_{i+1}^{1})$. We have $\bar{a}_{[m^{2}_{i-1}+1;m_{i}^{1};m_{i}^{1}+1;m^{1}_{i+1}]}$ and by Lemma 3.3

[TABLE]

which is a contradiction. 3. 3.

$(m_{i}^{2}<m_{i+1}^{1})$. We have by Lemma 3.3

[TABLE]

which is a contradiction.

The proof is completed.

$\Box$

Remark The stable partition can be obtained by considering the following simple deterministic dynamical system. We have $n$ particles starting from $x_{1}<x_{2}<\dots<x_{n}$. The $i^{\text{th}}$ particle has speed $a_{i}$. Each particle moves with a constant speed on the real line until it collides with one of its neighboring particle (if it happens). Then both the particles coalesce and from this time on they move with the proportional speed which is the mean of speed of colliding particles, and so on. Ultimately the particles will form never colliding groups, which are the same as in the stable partition of $\boldsymbol{a}$. Notice that resulted grouping do not depend on a starting position $\boldsymbol{x}$.

4 The theorem and examples.

We begin introducing some notations. Suppose that $\boldsymbol{a}$ has a stable partition with characteristics $q,(m_{i}),q^{\prime},(m^{\prime}_{i})$ respectively. In the sequel we will use the following notations:

[TABLE]

[TABLE]

[TABLE]

Moreover we define a function

[TABLE]

Remark that from Lemma 5.1 it will follow

[TABLE]

where

[TABLE]

Using this notation we now state a proposition which is useful for calculations in some cases.

Proposition 4.1

[TABLE]

Remark that formula (4.14) does not give us straightforward asymptotic because integral $I(\boldsymbol{a},t)$ depends on $t$. However in some cases this dependence vanishes and this is why Proposition 4.1 can be sometimes useful.

The next theorem gives us asymptotic for all cases.

Theorem 4.2

For some $C$ given below, as $t\rightarrow\infty$

[TABLE]

$\gamma$, $\alpha$, and $h(\boldsymbol{x})$ are defined in (4.10),(4.11),(4.12) respectively.

To show $C$ we need few more definitions. Let

[TABLE]

Define now

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

where $S_{kl}=(n-2)k$ for $k\leq l$ and $S_{kl}=S_{lk}$. In the remaining part of this section we diplay some special cases.

Example 2 $(a_{1}=a_{2}=\dots=a_{n})$ This is no drift case. Here $q=n$ and $m_{1}=1,m_{2}=2,\dots,m_{n}=n$, also $q^{\prime}=1$ and $m^{\prime}_{1}=n$. In result $f_{m_{1}}=a_{1},\dots,f_{m_{n}}=a_{n}$. Let $a$ be the common value of the drift. Using Proposition 4.1 we have

[TABLE]

First we notice that since all the coordinates in vector $\boldsymbol{f}$ are the same, we have

[TABLE]

Furthermore $W-\boldsymbol{f}\sqrt{t}=W$ because $y_{1}<y_{2}<\dots<y_{n}$ if and only if $y_{1}+a\sqrt{t}<y_{2}+a\sqrt{t}<\dots<y_{n}+a\sqrt{t}$. Finally we write

[TABLE]

where

[TABLE]

Before we state the next example we prove the following lemma.

Lemma 4.3

If $\boldsymbol{a}\in W$, then $\{W-\boldsymbol{a}t\}\rightarrow{\rm IR}^{n}$ as $t\rightarrow\infty$.

*Proof. * Let $\boldsymbol{a}\in W$. We show that for all ${\boldsymbol{y}\in{\rm IR}^{n}}$ there exists ${s>0}$, such that for all ${t>s}$, $\boldsymbol{y}\in\{W-\boldsymbol{a}t\}$. Let $y\in{\rm IR}^{n}$. We note $b_{i}=y_{i+1}-y_{i}$ and $d_{i}=a_{i+1}-a_{i}$. Condition $\boldsymbol{a}\in W$ implies $d_{i}>0$ for all $i=1,2,\dots,n-1$. We take $s={\max\{-b_{i},0\}}/{\min\{d_{i}\}}$ and $t>s$. Set $z_{i}=y_{i}+ta_{i}$, then we get that $\boldsymbol{z}\in W$, because

[TABLE]

Thus for $t>s$ we have $\boldsymbol{y}=\boldsymbol{z}-t\boldsymbol{a}$, where $\boldsymbol{z}\in W$, and so $\boldsymbol{y}\in\{W-ta\}$ for all $t>s$.

$\Box$

Example 3 $(a_{1}<a_{2}<\dots<a_{n})$ This is the case of non-colliding drifts. Here $q=q^{\prime}=n$, $m_{1}=m^{\prime}_{1}=1,\dots,m_{n}=m^{\prime}_{n}=n$, $f_{m_{1}}=a_{1},\dots,f_{m_{n}}=a_{n}$. Using Proposition 4.1 we have

[TABLE]

By Lemma 4.3 we have that

[TABLE]

Finally we write

[TABLE]

This result was derived earlier by Biane et al [2]

Example 4 Case when $q=q^{\prime}=1$. This is the case of a one irreducible drift vector. Here $m_{1}=m^{\prime}_{1}=n$, $f_{1}=f_{2}\dots=f_{n}=\bar{a}_{[1;n]}=\frac{a_{1}+\dots+a_{n}}{n}$. Using Proposition 4.1 we have

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We now analyze a remaining situation for $n=3$.

Example 5 $(a_{1}>a_{2}$ and $\frac{a_{1}+a_{2}}{2}<a_{3})$. This is the case of two subsequences. Thus $q=2,\ q^{\prime}=2$ and $m_{1}=m^{\prime}_{1}=2,\ m_{2}=m^{\prime}_{2}=3$. By Theorem 4.2 we have

[TABLE]

where

[TABLE]

5 Auxiliary results.

For the proof we need a set of lemmas and propositions, presented in subsections below.

5.1 Useful lemmas.

We need a few technical lemmas, which we state without proofs.

Lemma 5.1

For $\boldsymbol{a}\in{\rm IR}^{m}$

[TABLE]

Lemma 5.2

For $\boldsymbol{a},\boldsymbol{z}\in{\rm IR}^{m}$

[TABLE]

The proof of the following lemma follows easily from Lemmas 5.1 and 5.2.

Lemma 5.3

For $\boldsymbol{a},\boldsymbol{f}\in R^{n}$ such that $\boldsymbol{f}$ is is a vector obtained from the stable partition of $\boldsymbol{a}$, and $\boldsymbol{z}\in R^{n}$, we have

[TABLE]

Lemma 5.4

[TABLE]

then

[TABLE]

Note that $(\boldsymbol{A}^{-1})^{T}\boldsymbol{A}^{-1}$ is symmetric.

By Proposition 2.2 we have

[TABLE]

We now introduce new variable $\boldsymbol{z}$ by

[TABLE]

where $\boldsymbol{f}=(f_{1},\dots,f_{n})$ is a vector obtained from the stable partition of $\boldsymbol{a}$.

Finally we rewrite formula (2.5) in new variables by the use of Lemma 5.3:

Lemma 5.5

[TABLE]

5.2 Asymptotic behavior of determinant.

The following lemma is an extension of Lemma 2 from Puchała [9] .

We define functions

[TABLE]

for $\boldsymbol{k}=(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ and $0\leq k_{1}<\dots<k_{n}$ Functions $g$ corresponds to Schur functions $g_{\boldsymbol{k}}=s_{\boldsymbol{k}-(0,1,\dots,n)}$; see e.g. Macdonald [8], Ch. 1.3.

Lemma 5.6

Let $k_{0}=\sum_{j=1}^{q^{\prime}}\binom{\nu^{\prime}_{j}}{2}$

[TABLE]

where

[TABLE]

In particular as $t\rightarrow\infty$

[TABLE]

*Proof. *By $S_{n}$ we denote the group of permutations on $n$-set. We write

[TABLE]

Now the coefficient at $t^{-k/2}$ is equal to

[TABLE]

Recall that

[TABLE]

If $k_{i}=k_{j}$ and $f_{i}=f_{j}$, then the determinant $\det\left[e^{x_{i}f_{j}}x_{i}^{k_{j}}\right]$ is [math]. Thus we have non-zero determinant if $k_{i}$ are different for those $i$ such that $f_{i}$ are equal. Thus index $k$ such that $T_{k}$ is non-zero must be at least

[TABLE]

Moreover we get all nonzero $\det\left[e^{x_{i}f_{j}}x_{i}^{k_{j}}\right]$ putting in each subsequence

[TABLE]

all possible permutations of strictly ordered numbers from $\mathbb{Z}_{+}$ such that all sum up to $k$. Thus we have

[TABLE]

Again we notice that permutations in the determinant influence only by the change of sign. These signs and sums over the group of permutations form determinants, thus we have

[TABLE]

$\Box$

Remark. Using Itzykson–Zuber integral (see e.g. [6]) we can write

[TABLE]

where $\mu({\,{\rm d}}U)$ is (normalized) Haar measure on the unitary group $\boldsymbol{U}(n)$. Now letting $t\to\infty$,

[TABLE]

and

[TABLE]

Hence, as $t\to\infty$

[TABLE]

This is a less detailed version of the formula from Lemma 5.6.

6 Proof of the Theorem.

Using (5.19) and formula (5.18) we write

[TABLE]

First we will analyze above expression by taking only the first term in the sum (6.20), and then we show that it gives the right asymptotic. Thus the first term equals to

[TABLE]

where $I(\boldsymbol{a},t)$ was introduced in (4).

6.1 Asymptotic behavior of integral.

If $\boldsymbol{s}=\boldsymbol{A}\boldsymbol{z}$, where

[TABLE]

than $z_{u}-z_{v}=s_{v}+s_{v+1}+\dots+s_{u-1}$ and

[TABLE]

Hence by Lemma 5.4 we have

[TABLE]

where $\boldsymbol{s}_{(n)}$ is obtained from $\boldsymbol{s}$ by deleting the $n^{\text{th}}$ coordinate and $\boldsymbol{A}_{(n)}$ is matrix $\boldsymbol{A}$ without $n^{\text{th}}$ row and $n^{\text{th}}$ column.

After substitution $\boldsymbol{s}=\boldsymbol{A}\boldsymbol{z}$, integral $I(\boldsymbol{a},t)$ is

[TABLE]

It is important to notice that the second exponent in integral $I(\boldsymbol{a},t)$ in (6.1) depends only on those $s_{i}$, where $i\notin\{m_{1},\dots m_{q}\}$. We also see that if $m_{i-1}<k<m_{i}$, then the coefficient at $s_{k}$ in (6.1) is

[TABLE]

and it is strictly negative by the definition of the stable partition. Note also that polynomials $H$ in integral $I(\boldsymbol{a},t)$ depends only on $s_{j}$, where $j\notin\{m^{\prime}_{1},\dots,m^{\prime}_{{q^{\prime}}}\}$.

We now introduce new variables $\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{n-1})$ by

[TABLE]

We define function $K$ by $K\left(\boldsymbol{\xi}_{(m^{\prime}_{k-1},m^{\prime}_{k})},t\right)=H\left(\boldsymbol{s}_{(m^{\prime}_{k-1},m^{\prime}_{k})}\right)$.

Consider now $H\left(\boldsymbol{s}_{(1;m^{\prime}_{1})}\right)$. Since $\boldsymbol{m}^{\prime}$ is a subsequence of $\boldsymbol{m}$, we recall that $i_{1}$ is such that $m_{i_{1}}=m^{\prime}_{1}$. Similarly are defined $i_{1},\ldots,i_{q^{\prime}}$. We now factorize $H\left(\boldsymbol{s}_{(1;m^{\prime}_{1})}\right)$ into parts in which there in none of $m_{i}$, where is exactly one $m_{i}$, exactly two and so on. Thus

[TABLE]

We make analogous factorization for other $H(\boldsymbol{s}_{(m_{k-1};m_{k})})$.

Lemma 6.1

As $t\to\infty$

[TABLE]

*Proof. *After the substitution we get

[TABLE]

It is not difficult to see that asymptotic behavior of the above expression is

[TABLE]

In result the whole polynomial is asymptotically

[TABLE]

$\Box$

For substitution (6.25), we have ${\,{\rm d}}\boldsymbol{s}_{(n)}=t^{-(n-q)/2}{\,{\rm d}}\boldsymbol{\xi}$. Note that $f_{k+1}=f_{k}$ for $k\neq m_{i}$, and hence the integration on the $k^{\text{th}}$ coordinate starts from [math]. On the other hand if $k=m_{i}$ for some $i$, and $k\neq m^{\prime}_{j}$ for every $j$, then we also have $f_{k+1}=f_{k}$ and therefore the integration starts from [math]. Finally if $k=m_{i_{j}}$ for some $j$, then $f_{k+1}>f_{k}$ and the integrations starts from $(f_{k}-f_{k+1})\sqrt{t}$. Hence we have after the substitution

[TABLE]

So we can clearly see that $\prod_{k=1}^{q^{\prime}}K(\boldsymbol{\xi}_{(m^{\prime}_{k-1},m^{\prime}_{k})},t)$ depends only on $\xi_{i}$’s such that $i\notin\{m_{l_{1}},\dots,m_{l_{q^{\prime}}}\}$ and it can be factorized into a part which depends only on $i\notin\{m_{1},\dots,m_{q}\}$ and a part that depends on $i\in\{m_{1},\dots,m_{q}\}\setminus\{m_{l_{1}},\dots,m_{l_{q^{\prime}}}\}$. Thus finally we can write

[TABLE]

Hence

[TABLE]

Concluding we have

[TABLE]

where $C_{1}$ depends only on drift vector $\boldsymbol{a}$.

6.2 Proof of Theorem 4.2.

Following considerations of Section 6.1, notice first that it suffices to take the first term from the sum (6.20) for asymptotic analysis because next terms consists of positive rank polynomials of variable $\boldsymbol{z}$ and therefore they will tend to zero faster after substitution (6.25). For the proof of the main theorem we have to plug the asymptotics (6.26) to integral (6.20).