Asymptotic negligibility

2020 Mathematics Subject Classification: Primary: 60F99 [MSN][ZBL]

A property of random variables indicating that their individual contribution as components of a sum is small. This concept is important, for example, in the so-called triangular array. Let the random variables (; ) be mutually independent for each , and let

If for all and , at sufficiently large values of , the inequality

(1)

is satisfied, the individual terms are called asymptotically negligible (the variables then form a so-called zero triangular array). If condition (1) is met, one obtains the following important result: The class of limit distributions for ( are certain "centering" constants) coincides with the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution). If the distributions of converge to a limit distribution, , and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all and for all sufficiently large one has

(2)

then the following statement is valid: If (2) is met, the limit distribution for can only be a normal distribution (in particular with variance equal to zero, i.e. a degenerate distribution).

References

[a1]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966) pp. 210 MR0210154 Zbl 0138.10207