On the point process of near-record values.

Let $$(X_n)$$ be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution $$F$$ . An observation $$X_n$$ is a near-record if $$X_n\in (M_{n-1}-a,M_{n-1}]$$ , where $$M_{n}=\max \{X_1,\ldots ,X_{n}\}$$ and $$a>0$$ is a parameter. We analyze the point process $$\eta $$ on $$[0,\infty )$$ of near-record values from $$(X_n)$$ , showing that it is a Poisson cluster process. We derive the probability generating functional of $$\eta $$ and formulas for the expectation, variance and covariance of the counting variables $$\eta (A), A\subset [0,\infty )$$ . We also obtain strong convergence and asymptotic normality of $$\eta (t):=\eta ([0,t])$$ , as $$t\rightarrow \infty $$ , under mild tail-regularity conditions on $$F$$ . For heavy-tailed distributions, with square-integrable hazard function, we show that $$\eta (t)$$ grows to a finite random limit $$\eta (\infty )$$ and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data. [ABSTRACT FROM AUTHOR]