I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are mathematically rigorous results in this direction, or at least some clarifications would be helpful. What is non-typical in the example is that the "probability measure" is not positive, but rather complex valued (though still normalized by one).

The example is taken from the book "Gauge fields and strings", $\S$ 9.1, by A. Polyakov. The argument is a part of computation of some path integral.

Let us fix $T>0$. Divide the segment $[0,T]$ to $T/\varepsilon$ parts of equal length $\varepsilon$. For small $c>0$ consider the integral $$\int_{\mathbb{R}^{T/\varepsilon}}\left(\prod_{t=1}^{T/\varepsilon}d\gamma_t(\gamma_t-ic)^{-2}e^{i\varepsilon (\gamma_t-ic)}\right)\Phi(R;-i\varepsilon\sum_t(\gamma_t-ic)^{-1}),$$ where $\Phi(R,x)=x^{-2}\exp(-R^2/x)$. (Here $R$ is a real number; my notation is slightly different from the book.)

The measure is not normalized, but one can divide by the total measure. Clearly $(\gamma_t-ic)^{-1}$ are i.i.d. The above integral depends only on their sum $\sum_{t=1}^{T/\varepsilon}(\gamma_t-ic)^{-1}$. Thus formally it looks like one is in position to apply some form of LLN when $\varepsilon\to 0$ and replace this sum inside $\Phi$ by the expectation of $(\gamma_t-ic)^{-1}$ times $T/\varepsilon$. (In fact Polyakov gives few more estimates of the variance to justify that. It would be standard if the measure was positive, but otherwise it looks mysterious to me.)

  • $\begingroup$ Technically one can do that. But I think this is not what is done in the book mentioned above. $\endgroup$
    – makt
    Apr 26, 2015 at 10:04
  • 3
    $\begingroup$ I gave an answer at physicsoverflow.org/30561 $\endgroup$ Apr 28, 2015 at 11:50


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