On the point process of near-record values

Let $$(X_n)$$ ( X n ) be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution $$F$$ F . An observation $$X_n$$ X n is a near-record if $$X_n\in (M_{n-1}-a,M_{n-1}]$$ X n ∈ ( M n - 1 - a , M n - 1 ] , where $$M_{n}=\max \{X_1,\ldots ,X_{n}\}$$ M n = max { X 1 , … , X n } and $$a>0$$ a > 0 is a parameter. We analyze the point process $$\eta $$ η on $$[0,\infty )$$ [ 0 , ∞ ) of near-record values from $$(X_n)$$ ( X n ) , showing that it is a Poisson cluster process. We derive the probability generating functional of $$\eta $$ η and formulas for the expectation, variance and covariance of the counting variables $$\eta (A), A\subset [0,\infty )$$ η ( A ) , A ⊂ [ 0 , ∞ ) . We also obtain strong convergence and asymptotic normality of $$\eta (t):=\eta ([0,t])$$ η ( t ) : = η ( [ 0 , t ] ) , as $$t\rightarrow \infty $$ t → ∞ , under mild tail-regularity conditions on $$F$$ F . For heavy-tailed distributions, with square-integrable hazard function, we show that $$\eta (t)$$ η ( t ) grows to a finite random limit $$\eta (\infty )$$ η ( ∞ ) and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data. Copyright Sociedad de Estadística e Investigación Operativa 2015