Turbulence as a Problem in Non-equilibrium Statistical Mechanics

First, we obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails. Then we derive stable limit theorems for sums of the form (sum _{Ntge nge 1}Fbig (X_{q_1(n)},ldots ,X_{q_ell (n)}big )) where F is a polynomial, (q_i(n)) is either (n-1+i) or ni and (X_n,nge 0) is a sequence of independent identically distributed random variables with heavy tails. Our results can be viewed as an extension to the heavy tails case of the nonconventional functional central limit theorem from Kifer and Varadhan (Ann Probab 42:649–688, 2014).