2015/04/30 by Jaakko Lehtomaa, Lehtomaa, Jaakko
Business, Management and Accounting · Computer Science · Economics, Econometrics and Finance · Mathematics · #60E05 #60F05 #62E20 #Consumer Market Behavior and Pricing #Economic theories and models #FOS: Mathematics #Optimization and Search Problems #Probability (math.PR) #math.PR #msc:60E05 #msc:60F05 #msc:62E20
paper · pdf · doi:10.48550/arxiv.1504.08118
11 pages
arxiv created 2015/04/30 · openalex publication_date 2015/04/30 · arxiv updated 2015/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This note studies the asymptotic properties of the variable Zd:=(X1)/(d)|\X1+X2=d\, as d→ ∞. Here X1 and X2 are non-negative i.i.d. variables with a common twice differentiable density function f. General results concerning the distributional limits of Zd are discussed with various examples. Eventual log-convexity or log-concavity of f turns out to be the key ingredient that determines how the variable Zd behaves. As a consequence, two surprising discoveries are presented: Firstly, it is noted that the distributional limit is not strictly determined by the decay rate of the tail function. Secondly, it is shown that there exists a light-tailed distribution exhibiting behaviour that is commonly associated with heavy-tailed distributions i.e. the principle of a single big jump.