All Questions
8 questions
2
votes
1
answer
102
views
Approximation of $\Phi (p)$
I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
- 23
1
vote
2
answers
136
views
Uniform boundedness of integral?
I have perhaps a very simple question where I lack some inutition at the moment: Is the expression
$$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
- 536
2
votes
1
answer
403
views
Product of independent random variables and tail deconvolution
Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
- 205
8
votes
2
answers
891
views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
- 205
3
votes
1
answer
113
views
Asymptotic expansion of nonlinear Gaussian transformation in terms of covariance
I'm reading this paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "...
- 617
7
votes
5
answers
682
views
Bound on sum of complex summands involving binomial coefficients
I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|...
- 93
4
votes
1
answer
207
views
M-Wright function asymptotics
Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, $0<\nu<...
- 424
1
vote
1
answer
165
views
Estimate the scale of the power series with Poisson pdf/pmf-like terms
I would like to have an estimate for the series
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$
where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, $t$...
- 113