In the right-tail convention, every positive exponential moment is infinite. This is a tail property, not the assertion that the variance or all power moments must be infinite. A heavy-tailed subexponential distribution has the additional property that a sum's far tail is dominated by one large summand.
For nonnegative independent and identically distributed random variables with unbounded support, this is the displayed two-summand tail asymptotic. The probability that the maximum exceeds is , so the sum and maximum have asymptotically the same tail. This is the basic single-large-arrival effect relevant to queue overflow. The concentration-theory sub-exponential random variable uses a separate light-tail convention.
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A heavy-tailed distribution is a type of probability distribution that has a tail, which is the part of the distribution that represents extreme values, that is significantly heavier or more significant than that of the exponential distribution. This means that it has a higher probability of producing values far from the mean compared to lighter-tailed distributions, such as the normal distribution. In practical terms, this implies that heavy-tailed distributions can model phenomena where extreme events have a considerable chance of occurring.