Question

For which values of $\alpha \geq 0$ is the function $\exp(\log^\alpha(x))$ varying regularly?

score 203 · Answer 1 · 2026-04-25 15:55:24

203

Views

For which values of $\alpha \geq 0$ is the function $\exp(\log^\alpha(x))$ varying regularly?

Published on 25 Apr 2026 - 15:55

score 555 · Answer 2 · 2026-05-04 08:13:28

555

Views

regular and slowly varying functions examples $e^{\log(x)}$, $e^{\lfloor \log(x) \rfloor}$, $2 + \sin(x)$ and $e^{(\log(x))^\beta}$

Published on 04 May 2026 - 8:13

score 1k · Answer 3 · 2026-04-25 15:08:04

1k

Views

Slowly varying function without limit at infinity

Published on 25 Apr 2026 - 15:08

score 226 · Answer 4 · 2026-05-06 09:19:09

226

Views

Is a nondecreasing function bounded by a slowly varying function also slowly varying?

Published on 06 May 2026 - 9:19

score 50 · Answer 5 · 2026-04-27 18:58:31

50

Views

Slowly varying functions and the direction of convergence

Published on 27 Apr 2026 - 18:58

score 61 · Answer 6 · 2026-05-07 20:43:27

61

Views

Are Quasi-Polynomial Time Functions Regularly Varying?

Published on 07 May 2026 - 20:43

score 146 · Answer 7 · 2026-05-06 02:04:39

146

Views

Examples of Continuous Probability Distributions with Slowly Varying Upper Tail and Infinite Expectation

Published on 06 May 2026 - 2:04

score 65 · Answer 8 · 2026-05-07 20:30:07

65

Views

Prove for monotone $L \colon (0, \infty) \to (0,\infty)$, $L$ is slowly-varying if and only if there exists $x$ such that $\frac{L(tx)}{L(t)} \to 1$

Published on 07 May 2026 - 20:30

score 39 · Answer 9 · 2026-04-27 16:59:30

39

Views

Prove $\exp(\int^x_0 \frac{U(t)}{t} dt)$ is reguarly varying (Extreme Values, Regular Variation and Point Processes)

Published on 27 Apr 2026 - 16:59

score 21 · Answer 10 · 2026-05-05 13:31:58

21

Views

If $\lim_{n \to \infty} \lambda_n U(a_n x) \to \chi(x)$, $U$ is regulary varying and $\frac{\chi(x)}{\chi(1)} = x^\rho$

Published on 05 May 2026 - 13:31

score 82 · Answer 11 · 2026-05-05 20:34:37

82

Views

Slowlying varying function with no limit at infinity: using Karamata representation theorem

Published on 05 May 2026 - 20:34

score 31 · Answer 12 · 2026-05-05 13:15:56

31

Views

Reguarly varying function and convergence in probability

Published on 05 May 2026 - 13:15

score 32 · Answer 13 · 2026-04-25 14:56:23

Prove $V$ is slowly-varying if $\frac{V(tx)-V(t)}{a(t)} \to \log x$ (excercise 0.4.3.1 in Extreme Values, Regular Variation and Point Processes)

score 46 · Answer 14 · 2026-04-25 11:07:53

46

Views

Product of power-law distributed random variables

Published on 25 Apr 2026 - 11:07

score 478 · Answer 15 · 2026-04-28 13:17:30

478

Views

Composite of slowly varying functions

Published on 28 Apr 2026 - 13:17

List Question

For which values of $\alpha \geq 0$ is the function $\exp(\log^\alpha(x))$ varying regularly?

regular and slowly varying functions examples $e^{\log(x)}$, $e^{\lfloor \log(x) \rfloor}$, $2 + \sin(x)$ and $e^{(\log(x))^\beta}$

Slowly varying function without limit at infinity

Is a nondecreasing function bounded by a slowly varying function also slowly varying?

Slowly varying functions and the direction of convergence

Are Quasi-Polynomial Time Functions Regularly Varying?

Examples of Continuous Probability Distributions with Slowly Varying Upper Tail and Infinite Expectation

Prove for monotone $L \colon (0, \infty) \to (0,\infty)$, $L$ is slowly-varying if and only if there exists $x$ such that $\frac{L(tx)}{L(t)} \to 1$

Prove $\exp(\int^x_0 \frac{U(t)}{t} dt)$ is reguarly varying (Extreme Values, Regular Variation and Point Processes)

If $\lim_{n \to \infty} \lambda_n U(a_n x) \to \chi(x)$, $U$ is regulary varying and $\frac{\chi(x)}{\chi(1)} = x^\rho$

Slowlying varying function with no limit at infinity: using Karamata representation theorem

Reguarly varying function and convergence in probability

Prove $V$ is slowly-varying if $\frac{V(tx)-V(t)}{a(t)} \to \log x$ (excercise 0.4.3.1 in Extreme Values, Regular Variation and Point Processes)

Product of power-law distributed random variables

Composite of slowly varying functions

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