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Results tagged with asymptotics
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user 169474
Asymptotic behavior of functions, asymptotic series and related topics
2
votes
Accepted
Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicat...
Define $f$ on $]0,e^{-2}]$ by
$$f(x) = \frac{1}{-\sqrt{x}\ln(x)}.$$
Then $f$ is positive, decreasing since
$$\frac{\mathrm{d}}{\mathrm{d}x}\big(-\sqrt{x}\ln(x)\big) = \frac{-\ln(x)-2}{2\sqrt{x}} \ge 0 …
- 3,808
1
vote
Asymptotics of the unique root of a polynomial equation defined as a sum of rational express...
If the $\lambda_i$ take only two different values namely $\lambda_\max > \lambda_\min$, the equation $F(t)=s$ can be written
$$At^2(\lambda_\min+t)^2 + Bt^2(\lambda_\max+t)^2 = s (\lambda_\max+t)^2 (\ …
- 3,808
1
vote
Precise asymptotics for moments of order statistics of normal distribution
I call $f$ the density and $F$ the cumulative distribution function of $\mathcal{N}(0,1)$.
Since $X_{(n)} \ge X_{(n-1)}$,
$$X_{(n)}-X_{(n-1)} = \int_\mathbb{R} 1_{[X_{(n-1)} \le x < X_{(n)}]}dx.$$
Tak …
- 3,808