General Limit Distributions for Sums of Random Variables with a Matrix Product Representation

Abstract

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.

Appendix 1: A Periodic Irreducible Model

Periodic irreducible blocks $\fancyscript{B}(c)$ correspond to the case where all the loops of the digraph $\mathrm G(\fancyscript{B}(c))$ have a length which is a multiple of a base period $P>1$:

$$\begin{aligned} (\fancyscript{B}(c)^{k})_{ii} > 0 \iff k \in P \mathbb Z. \end{aligned}$$

(125)

Consequently, it is possible to partition the indices $\Xi _c$ in $P$ subsets $\Theta _{o}$ with $o \in \mathbb Z/P\mathbb Z$ such that the edges of $\mathrm G(\fancyscript{B}(c))$ only link indices from $\Theta _{o}$ to $\Theta _{o+1}$. The chain $\Gamma ^{|c}$ cycles over the set $\Theta _{o}$ with a period $P$ and therefore does not converge to a stationary state. However, our aim is not to obtain a convergence result for the chain $\Gamma ^{|c}$ but for the transition frequencies $\nu _{ij}$. The transition frequencies $\nu _{ij}$ are a global quantity that should not be influenced by the local periodic oscillation of $\Gamma ^{|c}$. In particular, we can consider $\Gamma ^{|c,o}$, the $P$ subchains obtained by jumping over a period

$$\begin{aligned} \Gamma ^{|c,o} = \left( \Gamma ^{|c}_{o}, \dots , \Gamma ^{|c}_{o+kP}, \dots , \Gamma ^{|c}_{o + P \lfloor (N t_c-o) /P \rfloor } \right) \!, \end{aligned}$$

(126)

where $\lfloor n\rfloor$ denotes the integer part of $n$. The chain $\Gamma ^{|c,o}$ corresponds to the Hidden Markov Chain of a matrix representation with structure matrix $\fancyscript{B}(c)^P$:

$$\begin{aligned} P(\Gamma ^{|c,o}) \!=\! \frac{ \fancyscript{L}\left( \fancyscript{B}(c)^{o-1} \left[ \prod _{k=1}^{\lfloor (N t_c-o)/P\rfloor } (\fancyscript{B}(c)^P)_{{\Gamma ^{|c,o}}_{k} {\Gamma ^{|c,o}}_{k+1}} \right] \fancyscript{B}(c)^{N t_c - P \lfloor (N t_c-o)/P\rfloor - o +1 } \right) }{ \fancyscript{L}\left( \fancyscript{B}(c)^{ N t_c } \right) }. \end{aligned}$$

(127)

Moreover, if we call $\nu \left( c,o\right)$ the transition frequencies of the subchain $\Gamma ^{|c,o}$ then

$$\begin{aligned} \nu (c) = \frac{1}{P} \sum _{o=1}^{P} \nu \left( c,o\right) . \end{aligned}$$

(128)

The structure matrix $\fancyscript{B}(c)^P$ is no longer periodic. If we relabel the indices of $\fancyscript{B}(c)$ in order to make the $\Theta _{o}$ contiguous, i.e. to ensure that $\Theta _{o} = {s_o, s_0+1, \dots , d_o-1, d_o}$, then the matrix $\fancyscript{B}(c)^P$ reads

$$\begin{aligned} \fancyscript{B}(c)^P = \left( \begin{array}{lll} D_{c,1} &{} &{} 0 \\ &{} \ddots &{} \\ 0 &{} &{} D_{c,P} \\ \end{array}\right) \end{aligned}$$

(129)

where $D_{c,o}$ are irreducible aperiodic square matrices of size $d_{c,o}$. The block diagonal structure of $\fancyscript{B}(c)^P$ derives from the fact that after $P$ jumps, the periodic chain $\Gamma ^{|c}$ goes back to its original set $\Theta _{o}$. For two indices $o\ne o'$, there cannot be any transition between $\Theta _{o}$ and $\Theta _{o'}$ in the matrix $\fancyscript{B}(c)^P$. In particular, if the final state of the subchain $\Gamma ^{|c}_{N t_c}=f$ belongs to the set $\Theta _{\omega }$ then for a non-zero probability subchain $\Gamma ^{|c}$, the chain $\Gamma ^{|c,o}$ stays inside the block $\Theta _{\omega +o-N t_c}$:

$$\begin{aligned} \forall k,\quad {\Gamma ^{|c,o}}_k \in \Theta _{\omega +o-N t_c}. \end{aligned}$$

(130)

Taking in account the property Eq. (130), Eq. (127) simplifies to

$$\begin{aligned} P(\Gamma ^{|c,o}) = \frac{ \fancyscript{L}\left( \fancyscript{B}(c)^{o-1} \left[ \prod _{k=1}^{\lfloor (N t_c-o)/P\rfloor } (D_{c,\omega +o-N t_c})_{{\Gamma ^{|c,o}}_{k} {\Gamma ^{|c,o}}_{k+1}} \right] \fancyscript{B}(c)^{N t_c - P \lfloor (N t_c-o)/P\rfloor - o +1 } \right) }{\fancyscript{L}\left( \fancyscript{B}(c)^{ N t_c } \right) }. \end{aligned}$$

(131)

The subchain $\Gamma ^{|c,o}$ therefore converges to the stationary state $p_{\mathrm {st}}(c, \omega +o-N t_c)$ associated with the structure matrix $D_{c, \omega +o-N t_c}$. As in the aperiodic case, the transition frequencies are therefore

$$\begin{aligned} \nu \left( c,o\right) _{ij} \overset{\text {a.s}}{\rightarrow } \frac{ \rho (c, \omega +o-N t_c)_i \, \fancyscript{B}(c)_{ij} \, \eta (c , \omega +o-N t_c)_j }{\Lambda } \end{aligned}$$

(132)

where $\rho (c, o)$ and $\eta (c,o)$ are respectively the left- and right-eigenvectors of the block $D_{c,o}$ (embedded in the whole vector space of $\fancyscript{B}(c)$). Combining Eqs. (128) and (132) yields

$$\begin{aligned} \nu (c) \overset{\text {a.s}}{\rightarrow } \frac{1}{P} \sum _{o=1}^{P} \frac{ \rho (c,o)_i \fancyscript{B}(c)_{ij} \eta (c,o)_j }{\Lambda }. \end{aligned}$$

(133)

The left and right eigenvectors of $\fancyscript{B}(c)$ associated with $\Lambda$, respectively $\eta (c)$ and $\rho (c)$, are exactly

$$\begin{aligned} \eta (c) = \sum _{o=1}^P \eta (c,o),\end{aligned}$$

(134)

$$\begin{aligned} \rho (c) = \sum _{o=1}^P \rho (c,o). \end{aligned}$$

(135)

Moreover, the support of the eigenvectors $\rho (c,o)$ and $\eta (c , o' )$ are disjoint if $o \ne o'$, consequently

$$\begin{aligned} \rho (c) \eta (c)^T = \frac{1}{P} \sum _{o, o'} \rho (c, o) \eta (c, o')^T = \frac{1}{P} \sum _{o=1}^P \rho (c,o)\eta (c,o)^T . \end{aligned}$$

(136)

Equation (133) therefore reads

$$\begin{aligned} \nu (c)_{i,j} \overset{\text {a.s}}{\rightarrow } \frac{ \rho (c)_i \fancyscript{B}(c)_{ij} \eta (c)_j}{\Lambda }. \end{aligned}$$

(137)

Hence the transition frequencies are exactly the same as the transition frequencies for the aperiodic case derived in Eq. (62).