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Heavy Tail Distribution - Weibull

I know that the Weibull distribution exhibits sub-exponential heavy tail behavior when the shape parameter is <1. I need to demonstrate this using the limiting definition of a heavy tail distribution:

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How to include the cumulative distribution function (CDF) or any other equation characteristic of the Weibull distribution to prove that this limit is satisfied?

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1 answer

CDF Weibull distribution 1 - exp(-(x/lambda)^k) = P(X <= x) .

So,

P(X > x) = 1 - CDF = exp(-(x/lambda)^k),

and

lim exp(lambda * x) * P(X > x) = lim exp(lambda x) * exp( - (x/lambda)^k)
                               = lim exp(lambda x - x^k/lambda^k)

k<1, x , lambda>0, lambda x , x^k/lambda^k ( ). , lambda x x^k/lambda^k. , lambda x - x^k/lambda^k .

, .

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