Probabilities of large values for sums of i.i.d. non-negative random variables with regular tail of index -1

2021/05/10 by Birkner, Matthias, Yuan, Linglong
#60E07 #60F10 #92D10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2105.04265

Abstract

Let ξ₁, ξ₂, … be i.i.d. non-negative random variables whose tail varies regularly with index -1, let S_n be the sum and M_n the largest of the first n values. We clarify for which sequences x_n→∞ we have \mathbb P(S_n ≥ x_n) ∼ \mathbb P(M_n ≥ x_n) as n→∞. Outside this regime, the typical size of S_n conditioned on exceeding x_n is not completely determined by the largest summand and we provide an appropriate correction term which involves the integrated tail of ξ₁.

Related

• A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums of independent identically distributed random variables
• A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums of independent identically distributed random variables
• Large Deviation Probabilities for Sums of Random Variables with Heavy or Subexponential Tails
• Large Deviations Of Sums Mainly Due To Just One Summand
• Contribution of the Extreme Term in the Sum of Samples with Regularly Varying Tail