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I interpret the question as follows (cf. the comment by Nate Eldredge). For each natural $n$, let $X_n,X_{n,1},\dots,X_{n,n}$ be independent identically distributed (i.i.d.) random variables (r.v.'s) …

Your question is unclear. E.g., if you allow $c$ to depend on the distribution of $X_1$, you can let $c=1/E|X_1|^3$. Then $cb=cnE|X_1|^3=n\ge1$, so that the inequality in question trivially holds. Y …

Take any real $p\ge0$. Assume that $E|X|^p+E|Y|^p<\infty$ and $0^0:=1$. For $p>2$, assume also that $EX=EY$. Let \begin{equation*} D_p:=2E|X-Y|^p-(E|X_1-X_2|^p+E|Y_1-Y_2|^p). \end{equation*} The …

The independence will be then in general lost. E.g., let $X_1,\dots,X_n$ be independent random variables each uniformly distributed on $[0,1]$. Let $M:=\max(X_1,\dots,X_n)=X_\nu$, so that $\nu$ is uni …

The problem is one of linear programming. To see this more clearly, let us assume for a moment that $Y$ is a finite set (rather than $\mathbb{R}^m$). Let me write $\mu f^{-1}$ instead of $f_\#\mu$. Th …

This is the result of me trying to prove the identity in Brendan McKay's answer. Consider, a bit more generally, any nonnegative independent r.v.'s $X$ and $Y$, still with $X_1,X_2$ being independent …

$Var\,f$ can be on the order of $n$ (but not more than that). Indeed, let $U$ and $N$ be independent random variables such that $P(U=1)=:p=1-P(U=0)=:q$ and $P(N=i)=1/n$ for all $i\in[n]:=\{1,\dots,n\} …

According to the multinomial probability mass function formula, the expected maximum frequency in $n$ rolls of a fair die is $$e_n=\frac1{6^n}\sum\frac{n!}{x_1!\cdots x_6!}\,\max(x_1,\dots,x_6),$$ whe …

There are such versions of the law of the iterated logarithm even for independent random vectors in an arbitrary separable Banach space. See e.g. Theorems 4.1 and 4.2. In the case when the Banach spac …

If you want all your conditions to hold for all real $x,y$, then that is impossible. Indeed, if $f(x,1-x)=\frac12$ and $f(0,x)=0$ for all real $x$, then $0=f(0,1)=f(0,1-0)=\frac12$, which is a contrad …

$\newcommand\G{\mathscr G}$For $j=1,2$, let $E_j$ and $V_j$ denote, respectively, the conditional expectation and the conditional variance given $\G_j$. Given that $\G_1\subset\G_2$, you want to show …

The following are necessary and sufficient conditions for the central limit theorem you want: (i) for $p\in[1,2]$: $$\int_0^1(EX(t)^2)^{p/2}\,dt<\infty\tag{1}$$ (see e.g. Giné, p. 147); (ii) for $p\in …

$\newcommand\ov\overline\newcommand\R{\mathbb R}$This is just an application of Tonelli's theorem. Indeed, let $X:=X_1$, $Y:=(Y_2,\dots,Y_n)$, $Y_i:=X_i$ for $i\in\ov{2,n}$, where $\ov{k,l}:=[k,l]\cap …

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\ …

$\newcommand{\ep}{\varepsilon}$ Without loss of generality (wlog), $a_1\ge a_2\ge\cdots$ and $a_n\to0$ as $n\to\infty$. Then a sufficient condition for $P(I=S^1)=1$ is that for some real $k>3/4$ and …