This week, we will start to work on multivariate models, and *non-independence*. The first idea to discuss non-independence will be to use the concept ofexchangeability. A sequence of random variable is said to be exchangeable if for all

Since this inequality should hold for all it comes that necessarily .

de Finetti (1931): Let be a sequence of random variables with values in . is exchangeable if and only if there exists a distribution function on such that

From the exchangeability condition, for any permutation of ,

Hewitt & Savage (1955): Let be a sequence of random variables with values in . is exchangeable if and only if there exists a measure on such that

Note that in the Gaussian case, where are i.i.d. random variables. To go further on exchangeability and related topics, see Aldous (1985) (see also here).

This construction can be used in credit risk, to model defaults in an homogeneous portfolio, see e.g. Frey (2001),

> proba=function(s,a,m,n){ + b=a/m-a + choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)* + dbeta(t,a,b)},lower=0,upper=1)$value + }

Based on that function, it is possible to plot the probability distribution over . In the upper corner is plotted the density of the Beta distribution.

> a=2 > m=.2 + n=10 + V=rep(NA,n+1) + for(i in 0:n){ + V[i+1]=proba(i,a,m,n)} > barplot(V,names.arg=0:10)

Those two theorems are extremely close,

**De Finetti’s theorem**: a random sequence of random variables is exchangeable if and only if ‘s are conditionnally independent, conditionnally on some random variable .

**Hewitt-Savage’s theorem**: a random sequence is exchangeable if and only if ‘s are conditionnally independent, conditionnally on some sigma-algebra

Olshen (1974), proposed an interesting discussion about those theorems, see also in the *Encyclopedia of Statistical Science*,

The subtle difference between those two theorem is also discussed in Freedman (1965)

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thanks Julyan for the comment on the Bayesian paradigm… about the last comment, indeed, exchangeability (infinite exchangeability) is a strong concept of positive dependence. Moshe Shaked introduced the PDM concept (positive dependent by mixture) in 1977, and it is one of the strongest concept of positive dependence… But I don’t think I’ll have time to discuss too much concepts of (multivariate) stochastic orderings, even if I plan to discuss definitions of positive dependence (and perhaps, also, negative dependence).

For the Bayesian paradigm justification: de Finetti’s representation theorem proves that under the exchangeability assumption for the observations, there exist a model and a prior distribution on its parameter, of which the observations are a random sample.

It’s interesting to see that an infinite exchangeable sequence necessarily has a positive covariance (not only a Gaussian sequence, as anyway the covariance is constant), and for a finite sequence, it is bounded below by -sigma^2/(n-1).