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This is to prove the conjecture \begin{equation*} x_n\sim\sqrt3\,n^{1/2} \tag{1}\label{1} \end{equation*} (as $n\to\infty$). (For all integers $n\ge1$,) we have \begin{equation*} h_n:=x_{n+1}- …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\th …

This equality is not true. E.g., if $f(u)=u$, $x=0$, and $a=1$, then the left-hand side of the equality is $1$, whereas its right-hand side is $2$. The OP has now switched the meaning of $r$ from rad …

Such an ideal does not exist. Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So, $$\m …

$\newcommand\bar\overline$ Letting $t:=\ln p$, we see that the limit in question is the limit of $$d(t):=\frac1t\Big(\sum_1^n x_j e^{tx_j}\Big/\sum_1^n e^{tx_j}-m_{e^t}\Big)$$ as $t\to0$. Next, lettin …

$\newcommand\ep\varepsilon\newcommand\de\delta$Note that $$\sum_{k=0}^n k^a\binom nk=n^a2^n E\Big(\frac{X_n}n\Big)^a, \tag{1}\label{1}$$ where $X_n$ is a binomial random variable with parameters $n,1/ …

Consider the substitutions \begin{equation*} x_n=n+1/2+y_n/n,\quad y_n=u_n+5/8. \end{equation*} Then $u_1=-9/8$ and \begin{equation*} u_{n+1}=f_n(u_n) \end{equation*} for $n\ge1$, where \begi …

There is no limit distribution at all for $a:=\alpha\in(0,1)$: for each $x\in(0,1)$, the relative frequency \begin{equation*} f_n:=f_n(x):=\frac1n\,\sum_{j=1}^{n-1}1(x_j<x) \end{equation*} will be …

Concerning homogeneous polynomials: Let $P(x,y)=\sum_{j=0}^n a_j x^j y^{n-j}$ be such a polynomial, of degree $n$ such that $C:=P^{-1}(\{c\})$ is a simple closed curve for all large enough $c>0$. If …

Perhaps the following example will do. It concerns the weak law of large numbers (LLN, established by Khinchin in 1929, https://www.encyclopediaofmath.org/index.php/Law_of_large_numbers) for independe …

$\newcommand\ep\varepsilon\newcommand\de\delta$ Let us show more: \begin{equation*} \frac{F_{\ep}(x)}{F_{\ep}((1/n))}\to0\tag{$*$} \end{equation*} (as $\ep\downarrow0$). Indeed, \begin{equation*} …

Without loss of generality, $R=1$. Let $Z_1,\ldots,Z_n$ be iid standard normal random variables (r.v.'s). Then \begin{equation} \sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\cdot …

$\newcommand{\ep}{\varepsilon}$To begin, note that \begin{equation*} P(B_n)=1-\prod_{k\ge n}P(X_k\le k-n) =p:=1-\prod_{j\ge0}(1-P(X>j)). \tag{1}\label{1} \end{equation*} Next, \begin{equation* …

Even if $X_n\to c$ in probability for some real constant $c$, it is not necessary that $P(Y\le X_n)\to P(Y\le c)$ -- you also need to require that $P(Y=c)=0$. More generally, if the limit of $X_n$ is …

One such condition is that $f$ be absolutely continuous in $[0,h)^2$ for some $h\in(0,1)$ -- so that $$f(x,y)+f(0,0)-f(x,0)-f(0,y)=\int_0^x du\,\int_0^y dv\,g(u,v)$$ for some function $g$ integrable o …