$ \newcommand{\Ha}{\mathcal H} \newcommand{\p}[1]{\left(#1\right)} \newcommand{\tr}{\mathrm{tr}} \newcommand{\vect}[1]{\underline{#1}} \newcommand{\mR}{\mathcal{R}} \newcommand{\Lf}{\mathcal{L}} \newcommand{\mE}{\mathcal{E}} \newcommand{\mG}{\mathcal{G}} \newcommand{\mA}{\mathcal{A}} \newcommand{\mP}{\mathcal{P}} \newcommand{\mQ}[1]{\mathcal{Q}\p{#1}} \newcommand{\fLf}[1]{ \frac{\Lf\p{#1} } {\Lf\p{\mE^n}}} \newcommand{\Esp}[1]{\left<#1\right>} \newcommand{\bra}[1]{\left<#1\right|} \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\as}[1]{ \overset{\mathrm{a.s}} {#1} } \newcommand{\conv}[1] {\underset{#1 \rightarrow +\infty} {\rightarrow} } \newcommand{\convAs}{ \as{\conv{n}} } \newcommand{\R}{\mathbb{R}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\Dist}[1]{\overset{\mathcal{D}}{#1}} \newcommand{\convD} {\Dist{ \conv{n} } } \newcommand{\Lc}{L_c} \newcommand{\Lg}{L_{\Gamma}} \newcommand{\lNormal}[2]{\mathcal{N}_{#1,#2}} \newcommand{\iid}{\mathrm{i.i.d.}} \newcommand{\cpath}{\mapsto} \newcommand{\wcon}{\leftrightarrow} \newcommand{\scon}{\leftrightarrows} \newcommand {\card} {\mathrm{card}} $

At equilibrium:

Microcanonical ensemble: $ p(x_1, \dots, x_n ) = \mathrm{cst} $
Canonical ensemble: $ p(x_1, \dots, x_n ) = e^{-\beta \Ha(x_1,\dots, x_n)} $

Out-of-equilibrium:

Constant flow of heat or particles
Dynamic description
Correlation
Stationary distribution?

A simple and famous out-of-equilibrium systems

Asymmetric: Particles only move from left to right
Exclusion : One particle by site
Creation rate $ \alpha $
Destruction rate $ \beta $

How do we describe the stationary distribution?

Matrix product ansatz (Derrida and al 1993)
matrix $ R(x) $
Long range correlation

Aim

Theoretical framework for stationary distribution with correlation
Inspired from the Matrix product ansatz

\[ p(x_1,\dots,x_n) = \fLf{ \mR(x_1) \dots \mR(x_n)} \]

linear form $\Lf$: $ \Lf(M) = \tr(\mA^T M) $
- $ \mA $: $ d \times d $ positive matrix
$ \mR(x) $: $ d \times d$ positive matrix function
- structure matrix \[ \mE = \int_\R \mR(x) dx \]
- probability density function matrix \[ \mP_{i,j} (x) = \mR(x) / \mE_{i,j} \]

$ d> 1 $: Commutativity $\implies$ Correlation

Product structure: $p(x_1,\dots,x_n) = \fLf{ \mR(x_1) \dots \mR(x_n)}$
Moment matrix: $ \mQ{q} = \int x^q R(x) dx $

\[ \Esp{X_k^p} = \frac{\Lf\p{\mE^{k-1} \mQ{p} \mE^{n-k}}}{\Lf\p{\mE^n} } \] \[ \Esp{X_k X_l} = \frac{\Lf\p{\mE^{k-1} \mQ{1} \mE^{l-k-1} \mQ{1} \mE^{n-l}}}{\Lf\p{\mE^n} } \] \[ \Esp{X_k X_l X_m} = \frac{\Lf\p{\mE^{k-1} \mQ{1} \mE^{l-k-1} \mQ{1} \mE^{m-l-1} \mQ{1} \mE^{n-m}}}{\Lf\p{\mE^n} } \] \[ \dots \] Translation invariance: \[ p(X_{k_1}=x_1,\dots, X_{k_l}= x_l ) = p(X_{c+k_1}=x_1,\dots, X_{c+k_l}= x_l ) \]

Sufficient condition

\[ [\mA^T,\mE] = \mA^T \mE - \mE \mA^T = 0 \] \[ \forall M,\quad \Lf(M\mE) = \Lf(\mE M) \]

$ p(X_k=x) = \fLf{\mR(x)\mE^{n-1}}$
$ p(X_k=x, X_l = y) = \fLf{\mR(x)\mE^{l-k-1}\mR(y)\mE^{n-|l-k|-1}} $
$ p(X_k=x,X_l=y, X_m = z ) = \fLf{\mR(x) \mE^{l-k-1}\mR(y)\mE^{m-l-1} \mR(z) \mE^{n-|m-k|-1}}$

\[ p(x_1,\dots,x_n) = \fLf{ \mR(x_1) \dots \mR(x_n)} \] How do we generate a random vector $X$ for a given triple $(\mA,\mE,\mP)$?

Expand the matrix product \[ p(x_1,\dots,x_n)= \frac 1 {\Lf\p{\mE^n} } \sum_{\Gamma \in \{1,\dots,d\}^{n+1}} \mA_{\Gamma_1,\Gamma_{n+1}} \mE_{\Gamma_1,\Gamma_2} \mP_{\Gamma_1,\Gamma_2}(x_1) \dots \mE_{\Gamma_n,\Gamma_{n+1}} \mP_{\Gamma_n,\Gamma_{n+1}}(x_n) \] \[ p(x_1,\dots,x_n) = \sum_{\Gamma} \kappa(\Gamma) P(\vect X|\Gamma) \]
$\Gamma$, hidden Markov chain

Markov chain

A correlated random process with short-term memory:

Characterized by

A initial distribution $p(\Gamma_0)$
A transition probability $p_k( i \rightarrow j ) \equiv P(\Gamma_k=j | \Gamma_{k-1}=i )$

Markov chain $\Gamma$
Observable $\vect X= X_1, \dots, X_k$
$( X_k | \Gamma_k) $ is distributed according to the pdf $ p( X_k | \Gamma_k) $

Hidden Markov Chain $\Gamma$

\[ p( \Gamma_k=i | \Gamma_{k+1}=j, \Gamma_{n}=f ) = \frac{\p{\mE^{n-k}}_{j, f}}{ \p{ \mE^{n-k+1}}_{i,f}} \] \[ p(\Gamma_0=i,\Gamma_n=j) = \frac{A_{i,j} \mE^n_{i,j} }{\Lf\p{\mE^n}} \]

Conditional pdf $(X|\Gamma)$

\[ p( X_k=x | \Gamma_k=i, \Gamma_{k+1}=j) = \mP_{i,j}(x) \]

Non-homogeneous markov chain
Specific non-homogeneous Hidden Markov model:
- Hidden Markov Model $\nRightarrow$ Matrix representation

Matrix representation

Algebraic properties
Statistical properties computation

Hidden Markov Model

$2$-layer model: correlated layer $+$ independant layer
Efficient synthesis

\[ \Esp{X_k X_l} = \fLf{ \mE^{k-1} \mQ{1} \mE^{l-k-1} \mQ{1} \mE^{n-l} } \]

The dependency structure of $X$ depends on the behavior of $\mE^n$
$\lambda_k$ eigenvalues of $\mE$ ordered by their real parts $\Re (\lambda_1) \Re (\lambda_2) > \dots > \lambda_r$
$J_{k,l}$ Jordan block associated with eigeivalue $\lambda_k$

\[ \mE = B^{-1} \begin{pmatrix} J_{1,1} & 0 & \cdots & \cdots & 0 \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & J_{k,l} & \ddots& \vdots \\ \vdots & & \ddots & \ddots& 0 \\ 0 & \cdots & \cdots & 0 & J_{r,m} \end{pmatrix} B, \quad J_{k,l} = \begin{pmatrix} \lambda_k& 1 & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & \lambda_k \end{pmatrix} \]

Case 1: Short-range correlation

More than one distinct eigenvalue $\lambda_k$: Short-range exponential correlation
- $\Esp{X_k X_l} - \Esp{X_k}\Esp{X_l} \approx \sum \alpha_k \exp\p{- |l-k| [ \frac{1}{\tau_k} + 2\imath \pi \omega_k ] }$
- $\tau_k = 1/(\ln \lambda_1 - \ln |\lambda_k|)$
- $\omega_k = \mathrm {Arg}(\lambda_k)$

Case 2: Constant correlation

More than one block $J_{1,k}$: Constant correlation term

Case 3: Long-range correlation

At least one block $J_{1,k}$ with size $p > 1$: Polynomial long range correlation
- $\Esp{X_k X_l} - \Esp{X_k}\Esp{X_l} \approx P(\frac{k}{n}, \frac{k-l}{n}, \frac{l}{n} ), \quad P \in \R[X,Y,Z]$

$\mE$ irreducible $\iff \Gamma$ ergodic Irreducible matrix $\mE \iff G(\mE) $ is strongly connected
Short-range correlation

Disconnected components:
\[ \mE = \begin{pmatrix} 1 & & 0 \\ & \ddots & \\ 0& & 1 \end{pmatrix} \]
The chain $\Gamma$ is trapped inside its starting state
Constant correlation:
- $ \Esp{X_k X_l }- \Esp{X_k}\Esp{X_l} = \frac{\Lf\p{\mQ{1}^2} - \Lf\p{\mQ{1}}^2}{\Lf\p{\mE}}$

Irreversible transitions:
\[ \mE = \begin{pmatrix} 1 & \epsilon & 0 \\ & \ddots & \ddots & \\ 0 & & 1 \\ \end{pmatrix} \]
The chain $\Gamma$ can only stay in its current state or jump to the next
All chains with a non-zero probability and the same starting and ending points are equiprobable
Polynomial correlation: \[ \Esp{X_k X_l } \approx \sum_{r+s+t=d-1} c_{r,s,t} \p{\frac k n}^{r} \p{\frac {l-k} n}^{s} \p{1- \frac l n}^{t} \]

\[ \mE = \begin{pmatrix} I_1 & * & T_{k,l} \\ & \ddots & * \\ 0 & & I_r \\ \end{pmatrix} \]

Irreducible blocks $I_k$
Irreversibles transitions $T_{k,l}$
Correlation: Mixture of short-range, constant and long-range correlations

Short-range correlation $\implies$ Strongly connected component of size $s > 1$
Constant correlation $\implies$ More than one weakly connected component
Polynomial correlation $\implies$ More than one strongly connected component

Necessary but non sufficient conditions

Generation of random vector with prescribed correlation and marginal distribution:
- Matrix representation: Choice of $(\mA,\mE,\mP)$
- Hidden Markov Model: Numerical generation
Higher-order dependency structure: correlation of squares

Realization	Marginal	Correlation	Square corr.

Prescribed

Sum

\[ S(\vect X) = \frac{1} {n} \sum_{i=1}^n X_i \] Correlated random variables

Law of large numbers?
Central limit theorem?
Large deviations?

Two paths:

Hidden Markov chain representation
Matrix representation

$\nu_{i,j}$ fraction of $(i\rightarrow j)$-transition: \[\vect{\nu} = \p{ \frac{ \card\{ k / \Gamma_k= i, \Gamma_{k+1}= j\}}{n} }_{i,j} \]
$S(\vect X|\Gamma)$: sum of sums of $\iid$ random variables: \[ S(\vect X|\Gamma) = \sum_{i,j} \sum_{k=1}^{n \nu_{i,j}} (X_k | i, j ) \equiv S(\vect X | \vect \nu) \]
Standard convergence theorem (law of large numbers or central limit theorem )

\[ p( S(\vect X) = s ) = \sum_{\vect \nu} p(\vect \nu) p( S(\vect X | \vect \nu ) = s ) \]

Limit distribution for $S(\vect X|\vect \nu)$

\[ p( S(\vect X | \vect \nu ) = s ) \]

Limit distribution for $\vect \nu$

\[ p( \vect \nu ) \]

\[ \Downarrow \]

Limit distribution for $S(\vect X)$

\[ p( S(\vect X) = s ) \]

How $\vect \nu$ is distributed?
Difficulty: $\Gamma$ non-homogeneous Markov chain.

Three important subclasses:

Irreducible $\mE$ (short-range correlation)
- $\nu$ converges towards a dirac distribution
Identity $\mE$ (constant correlation)
- $\nu$ converges towards a discrete mixture of dirac distributions
Linear irreversible $\mE$ (long-range polynomial correlation)
- $\nu$ converges towards a uniform distribution on a $d$-simplex

Irreducible $\mE$ (short-range correlations):
- Standard limit laws
  
  + $\implies$
Identity $\mE$ (constant correlation):
- Discrete mixture of standard limit laws:
  
  + $\implies$
Linear irreversible $\mE$ (long-range correlation):
- Continuous mixture of limit laws
  
  + $\implies$

Combinations of three core behaviors
- Irreducible blocks: Fast convergence to the stationary state : dirac distribution
- Separated componentes : discrete mixture
- Irreversible transitions: continuous mixture
Limit laws :
- Discrete mixture of continuous mixture of standard limits distributions

Sum

\[ S(\vect X) = \frac{1} {n} \sum_{i=1}^n X_i \]

Law of large number: existence of a concentration point
Central limit theorem: fluctuations around this concentration point
Large deviation: fluctuations far away of the concentrations points

Large deviation principle

\[ P( S(\vect X)/n = s ) \approx e^{-n I(s) } \] Do we have a large principle?

Generating function \[ g_n(w) = \Esp{e^{w S_n}} \]
Scaled cumulant generating function $\Phi(w)$ \[ \Phi(w) = \lim_{n \rightarrow \infty} \frac{\ln g_n(w)}{n} \]

Gärtner-Ellis Theorem

If $\Phi(w)$ exists and is differentiable, $I(s) $ exists and is \[ I(s) = \sup_{w} \{w s - \Phi(w) \} \]

$\Phi(w) = g(w)$
$g(w)$ cumulant generating function

Large deviation principle:

\[ I(s) = \sup_{w} \{w s - g(w) \} \]

$\Phi(w) = \ln \lambda_1(w)$
$\lambda_1(w)$ dominant eigenvalue of $\mG(w)$
matrix generating function: \[ \mG(w) = \int_\R \mR(x) e^{-w x} dx \]

Large deviation principle for short-range correlation:

Short-range correlation: $\lambda_1(w)$ is differentiable near $0$

\[ I(s) = \sup_{w} \{w s - \ln \lambda_1(w) \} \]

Long-range or constant correlation

$\Phi(w)$ is not differentiable in $0$.

Constant correlation: Non-convex rate function
Polynomial correlation: Rate function with a flat branch