Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and $$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$ a stochastic matrix (ie a matrix whose elements are non negative and such that each row sum is one). Let $P^{\pi}_{\sigma} $ be the Markov probability measure associated with this matrix and with initial state $\sigma \in \Sigma$, ie $$P^{\pi}_{\sigma}(Y_1 = y_1, \ldots, Y_n = y_n) = \pi(\sigma, y_1) \Pi_{i=1}^{n-1} \pi(y_i, y_{i+1}). $$ Let $ h: \Sigma^{\mathbb{N}} \times \Sigma^{\mathbb{N}} \rightarrow \mathbb{R}$ be a deterministic function (here $\Sigma^{\mathbb{N}}$ is the space of all sequences comprising of elements of $\Sigma$). Consider the random variable $Z_{n}: \Sigma^{\mathbb{N}} \rightarrow \mathbb{R} $ given by $$ Z_{n}(Y) := \Sigma_{i=1}^{n} \Sigma_{j=i+1}^{n} \frac{h(\tau^{i}Y, \tau^{j}Y)}{n^2} $$ where $\tau_i$ is the shift operator, ie $$\tau^{i}(Y_1 Y_2 \ldots Y_i Y_{i+1} \ldots) := Y_{i+1} Y_{i+2} \ldots $$

My question is the following: under what conditions on $h$ can we say that the sequence of random variables $Z_n$ satisfy a "Large Deviation Principle"? Secondly, is there any way to compute the Rate function?

Of course, here we are looking at $\Sigma^{\mathbb{N}} $ as a probability space with the measure induced by the matrix $\Pi$.

I am aware that there are results known for additive functionals as given in chapter 3 of the book "Large Deviations: Technigues and Applications" by Amir Dembo and Ofer Zeitouni. I am wondering if something similar is known for the type of random variable I have considered.


1 Answer 1


Your function is a continuous functional $F$ of the empirical process (up to an exponentially negligible error, it is the function $F(\mu)= \int\int h(x,y) \mu(dx)\mu(dy)$). Now apply the contraction principle to get the rate function $I(x)=\inf\{J(\nu): F(\nu)=x\}$. Here J is the rate function for the empirical process (which is the specific entropy).

The needed condition is that needed to make $F$ continuous wrt weak topology. In particular, the influence of far coordinates on $h$ should decay.

You should look in chapter 6.5, not chapter 3, of DZ.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.