Approximation of the tail probability of randomly weighted sums and applications

作者：Zhang Yi; Shen Xinmei; Weng Chengguo^*

来源：Stochastic Processes and Their Applications, 2009, 119(2): 655-675.

DOI：10.1016/j.spa.2008.03.004

• Randomly weighted sums
• Asymptotics
• Regular variation
• Upper tail dependence
• Ruin probability
• Stochastic difference equations

Consider the problem of approximating the tail probability of randomly weighted sums Sigma(n)(i=1) theta(i)X(i) and their maxima, where {X(i), i >= 1} is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation and (theta(i), i >= 1} is a sequence of nonnegative random variables of {X(i), i >= 1} and satisfying certain moment conditions. Under the assumption that {X(i), i >= 1} has no bivariate upper tail dependence along with some other mild conditions,
Pr(max(1 <= k <= n) Sigma(k)(i=1) theta(i)X(i) > x) similar to Pr(Sigma(n)(i=1) Pr(theta(i)X(i) > x) similar to Sigma(n)(i=1) Pr(theta(i)X(i) > x),
and
Pr(max(1 <= k <=infinity) Sigma(k)(i=1) theta(i)X(i) > x) similar to Pr(Sigma(infinity)(i=1) Pr(theta(i)X(i)( ) > x) similar to Sigma(infinity)(i=1) Pr(theta(i)X(i) > x)
as x -> infinity. In doing so, no assumption is made on the dependence structure of the sequence {theta(i), i >= 1}.

• 出版日期2009-2
• 单位浙江大学