# Question about a special test of independence / autocorrelation

The goal is to design a full, true test of independence for a sequence of multivariate observations $$X_1,X_2,\dots,X_n$$, that is, testing if the observations (each assuming to come from a same distribution) are jointly independently distributed. The data may be coming from dynamical systems, the variance and/or expectation may be infinite, and as result the autocorrelations may not exist. This has nothing to do with $$\chi^2$$ tests or testing if autocorrelations are zero.

However, I will only discuss here the most basic case. I am interested to know if the statistic of the test, in the most basic case, is asymptotically normal and does not depend on the distribution of the $$X_k$$'s. Assume the $$X_k$$'s are univariate and we want to test if $$X_k$$ is independent of $$X_{k+1}$$.

So here's the test. You compute the empirical joint probabilities $$p_{\alpha,\beta}=P(X_k\leq\alpha, X_{k+1}\leq\beta)$$ for various values of $$0\leq\alpha,\beta\leq1$$. You also compute the empirical marginal probabilities $$p_\alpha=P(X_k\leq \alpha)$$. The statistic of the test is simply $$D=\sigma\sum_{\alpha,\beta}|p_{\alpha,\beta}-p_\alpha p_\beta|$$

where $$\sigma$$ is a normalizing constant. Under which conditions is $$D$$ asymptotically normal under $$H_0$$, that is in case of true independence? How to choose $$\sigma$$ and how to choose the $$\alpha,\beta$$ to have a sound test? Are there already any test achieving the same purpose? For simplicity, we can assume that $$X_k\in[0,1]$$, and the number $$n$$ of observations is large, say $$n=10^6$$.

Update

Let $$N$$ be the number of vectors $$(\alpha,\beta)$$ for which $$p_{\alpha,\beta}$$ is computed. In my case $$N\approx 4000$$, the $$\alpha,\beta$$ are evenly sampled in $$[0,1]$$, and $$X_k\in [0,1]$$. It makes sense to use $$\sigma$$ proportional to $$N$$. Note that the $$p_{\alpha,\beta}$$ are not independent. This is the price to pay for not using bins (some of which could be very sparse) together with a standard $$\chi^2$$ test. The theoretical distribution of $$D$$ can be approximated by simulation (resampling-like methods). It still should be asymptotically normal. Maybe a better statistic is

$$D^*=\sup_{\alpha,\beta}|p_{\alpha,\beta}-p_\alpha p_\beta|.$$

This makes the test more like a Kolmogorov-Smirnov one, rather than $$\chi^2$$-like. Also, in my case, I have the ability to produce as many samples $$X_1,\cdots,X_n$$ as I want, with $$n$$ as large as I want. One example is to start with $$X_1$$ a random deviate in $$[0,1]$$, and $$X_{k+1}=bX_k - \lfloor b X_k\rfloor$$, where the brackets represent the integer part function and $$b>1$$ is an integer. In that case, it is well known that the theoretical lag-$$m$$ autocorrelation in the infinite sequence $$(X_k)$$ is equal to $$1/b^m$$, and thus close to zero for large values of $$b$$. Also, the theoretical value of $$p_\alpha$$ is $$p_\alpha=\alpha$$. I am not interested in that case because all the theory is known, but in more complicated dynamical systems.

• In the example cited in the 'Update' section, computational errors for $X_k$ propagate exponentially fast, but the system is ergodic, so this is not a concern. Mar 28 at 2:14