scholarly journals A form of regular variation and its application to the domain of attraction of the double exponential distribution

1971 ◽  
Vol 17 (3) ◽  
pp. 241-258 ◽  
Author(s):  
Laurens Haan
Keyword(s):  
Exponential Distribution ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Double Exponential Distribution ◽  
Double Exponential

Extremes for the minimal spanning tree on normally distributed points

1998 ◽  
Vol 30 (03) ◽  
pp. 628-639 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Normal Distribution ◽  
Exponential Distribution ◽  
Spanning Tree ◽  
Dimensional Space ◽  
Edge Length ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
Union Of Balls

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let M n be the longest edge-length of the minimal spanning tree on these points; equivalently let M n be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2 M n - b n converges weakly to the Gumbel (double exponential) distribution, where b n are explicit constants with b n ~ (ν - 1)log log n. We also show the same result holds if M n is the longest edge-length for the nearest neighbour graph on the points.

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