[R] Validity of Pearson's Chi-Square for Large Tables

[dsimcha]

Is Pearson's Chi-Square test for contingency tables asymptotically unbiased
for large tables (large degrees of freedom) regardless of the expected
values in each cell?  The rule of thumb is that Pearson's Chi-square should
not be used when large numbers of cells have expected values < 5.  However,
I compared the results on 4x4 contingency tables for R's chisq.test using
chi-square approximation vs. chisq.test using a large number of monte carlo
simulations, and the results agree within a fairly small error.  This is
true even when every cell of the table has an expected value < 2.  I tried
several tables, but the best example was:

4  1  1  1
1  4  1  1
1  1  4  1
1  1  1  4

As expected, the chi-square approximation appears to be very poor when both
the expected values and degrees of freedom are small.  Is there a good
theoretical reason why the chi-square test seems to perform well on large
contingency tables even with small expected values?  Are the standard rules
of thumb overly simplistic?
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