View source for Asymptotic negligibility

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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|triangular array]]. Let the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137201.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137202.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137203.png" />) be mutually independent for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137204.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137205.png" /></td> </tr></table>

If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137207.png" />, at sufficiently large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137208.png" />, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

is satisfied, the individual terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372010.png" /> are called asymptotically negligible (the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372013.png" /> are certain  "centering"  constants) coincides with the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). If the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372014.png" /> converge to a limit distribution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372015.png" />, and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372017.png" /> for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372018.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

then the following statement is valid: If (2) is met, the limit distribution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372020.png" /> can only be a [[Normal distribution|normal distribution]] (in particular with variance equal to zero, i.e. a degenerate distribution).



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley  (1966)  pp. 210</TD></TR></table>