  Stretched exponential probability density functions (pdf), having the form of
the exponential of minus a fractional power of the argument, are commonly found
in turbulence and other areas. They can arise because of an underlying random
multiplicative process. For this, a theory of extreme deviations is developed,
devoted to the far tail of the pdf of the sum $X$ of a finite number $n$ of
independent random variables with a common pdf $e^{-f(x)}$. The function $f(x)$
is chosen (i) such that the pdf is normalized and (ii) with a strong convexity
condition that $f''(x)>0$ and that $x^2f''(x)\to +\infty$ for $|x|\to\infty$.
Additional technical conditions ensure the control of the variations of
$f''(x)$. The tail behavior of the sum comes then mostly from individual
variables in the sum all close to $X/n$ and the tail of the pdf is $\sim
e^{-nf(X/n)}$. This theory is then applied to products of independent random
variables, such that their logarithms are in the above class, yielding usually
stretched exponential tails. An application to fragmentation is developed and
compared to data from fault gouges. The pdf by mass is obtained as a weighted
superposition of stretched exponentials, reflecting the coexistence of
different fragmentation generations. For sizes near and above the peak size,
the pdf is approximately log-normal, while it is a power law for the smaller
fragments, with an exponent which is a decreasing function of the peak fragment
size. The anomalous relaxation of glasses can also be rationalized using our
result together with a simple multiplicative model of local atom
configurations. Finally, we indicate the possible relevance to the distribution
of small-scale velocity increments in turbulent flow.
