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The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows pr …

Define the smooth map $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) := \frac{1-\cos(x)}{x^2} = -\sum\limits_{k=1}^\infty \frac{(-1)^k}{(2k)!} x^{2k-2}$. I am looking for a nice bound on $|f^{(n)}(x …

For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions. Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{ …

I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \inf …

Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation \begin{align*} & f(x,y) = \frac{1}{1-y} + A_+( …

Long story short, I only get $C(n,f,\varepsilon) \, a^{-n (k-\varepsilon n-\frac{1}{2})}$ as a bound for $C^k$-functions for any $\varepsilon > 0$. I get a nice bound for every $f$ with absolutely su …

Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, …

Using the Radon-Nikodym Theorem this problem can be reduced to the case for a fixed measure. Suppose none of the $\mu_n$ is the zero-measure, i.e. $\mu_n(X)>0$ for all $n \in \mathbb{N}$. For a series …