Unbounded - crowlogic/arb4j GitHub Wiki

Yes, it is possible for a process (or a random variable) to be unbounded (i.e., it can take values up to infinity) and yet have a finite variance.

Variance is a measure of the spread or dispersion of the values that a random variable can take. In mathematical terms, the variance of a random variable (X) is defined as the expected value of the squared deviation from the mean, i.e.,

[ Var(X) = E[(X - E[X])^2] ]

Even if a random variable can take on arbitrarily large values, it can still have a finite variance if the probability of taking on those large values is sufficiently small. This is the case, for instance, for the normal distribution and the exponential distribution, both of which are unbounded (the normal distribution is unbounded on both sides, and the exponential distribution is unbounded on the positive side), but have finite variance.

However, there are also unbounded distributions that have infinite variance, such as the Cauchy distribution. So, the fact that a distribution is unbounded does not automatically tell you whether its variance is finite or not - this depends on the specific details of the distribution.

It's also worth noting that the maximum value of a random process is a slightly different concept, because a random process is a collection of random variables indexed by time or space, and each of these random variables can have its own distribution with its own maximum value. But the basic idea is the same: even if these maximum values are unbounded, the process can still have finite variance if the probability of taking on large values is sufficiently small at each point in time or space.