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Random numbers and other random objects
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In probability theory, a random number is just a random variable x,
i.e., a measurable function on the set Omega of possible experiments, 
that assigns to each experiment omega in Omega the value x(omega) 
of x in this experiment. Here Omega comes equipped with a normalized 
positive measure mu that assigns to every measurable subset S of Omega 
the probability mu(S) that a randomly chosen experiment belongs to S.

In the important, 'noninformative' case where the measure is invariant 
under a group transitive on Omega, so that all experiments are 
identical copies of one another, physicists refer to this set Omega 
as a (classical) 'ensemble', although they are usually too vague to 
express this in formal terms.
The terminology easily extends to the inhomogeneous case if one
allows in ensembles each realization with a different frequency.

Mathematicians prefer to leave the set Omega (which they call the 
'sample space') unspecified and talk about 'realizations' in place of 
'experiments'. Thus, for each experiment omega in Omega, x(omega) is 
a realization of x, i.e., what physicists would call the value found 
in this particular experiment. 

By giving a specific definition of the sigma algebra of interest
(defining which subsets of Omega are measurable), the measure assigning
probabilities, and a specific recipe defining the x(omega), one has a 
model world in which realizations make perfect sense. 

A difficulty is, of course, that we do not have such a model for the 
real world, and hence must resort to empirical approximations when 
treating real-life problems. (This places physicists at a slight 
disatvantage; however, there is the compensating advantage that their 
results apply to real life instead of only satisfying one's sense of 
beauty and precision....)

The only thing not specified in probability theory (unless one specifies
a particular model as indicated above) is the mechanism that draws 
the number, and hence there is no way to know which experiment omega 
has been realized. Therefore, probability theory makes only statements 
about _all_ realizations simultaneously.

Example. Given the axioms of probability theory,  a random number 
uniformly distributed between zero and one is defined as a random 
variable x such that
   <f(x)> = integral_0^1 f(s) ds
for all Lebesgue-integrable functions f on [0,1], and any x(omega) is a 
realization of it, i.e., an actual number in [0,1]. (In particular, 
random numbers are _not_ numbers; only their realizations are!)

Mechanisms to draw numbers that may be used as approximations to a 
sequence of independent realizations x(omega) are called randon number 
generators. They do not produce random numbers (since random numbers 
are not numbers but measurable functions). Instead, they produce 
sequences that look like typical realizations of sequences of 
independent, uniformly distributed random numbers - in the sense that 
they usually pass with high confidence level certain statistical tests 
valid for such random sequences (this is discussedin any textbook on 
statistics).
Therefore, the numbers they generate are used in practice as (often 
completely adequate) substitutes for random numbers. 

(On the other hand, there is no uniformly distributed random natural 
number since the uniform measure on natural numbers, 
mu(f) = sum_{k>=0} f(k) is not normalizable.)


Random numbers are comparably simple objects. More complicated random 
objects need more sophisiticated ensembles but otherwise everything 
remains analogous.

Let us consider the physically important example of Brownian motion.
Brownian motion (the random walk in space) is modelled by an ensemble 
whose realizations (members) are the H"older differentiable 
functions on R^3 with exponent 1/2. The probability of any particular 
realization of a random walk is exactly zero, and statements with
positive probability must hold in uncountably many realizations.
Nevertheless, the ensemble is precisely the set Omega composed of all 
such realizations. And the appropriate sigma algebra carrying the 
Wiener measure needed to describe the random walk is indeed an algebra 
of subsets of Omega.

Repeatedly tossing a fair coin is also a (kind of trivial) stochastic
process. A fair coin that can be thrown an unlimited number of times 
with independent outcomes (sampling with replacement) cannot be 
modelled by the sigma algebra 2^{0,1} over Omega_1 ={0,1}, since 
this has not even two independent bits. Its sigma algebra is based 
on the infinite ensemble Omega_inf consisting of all possible 
sequences of outcomes, and is the tensor product of infinitely many 
copies of 2^{0,1}. This setting is necessary in order to provide 
meaning to the concept of 'independent trial' which 
underlies most of statisitcal reasoning.

Because of the assumed independence of the trials, one can reduce all 
computations to computations within 2^{0,1}. This is generally done
in elementary probability theory, to simplify the presentation.
But once one looks at binary processes which are even slightly 
correlated (history-dependent), one needs the full sigma algebra 
over Omega_inf.