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You may want to model the situation by assuming that the alternative distribution is the mixture $P_t:=(1-t)P_0+tP_1$ of the distributions $P_0$ and $P_1$, for some $t\in(0,1]$. You may then want to t …

Let $$S_n=\sum_{k=1}^n P(A_k),\quad T_n:=\sum_{1\le i<k\le n}P(A_i)P(A_k),$$ $$R_n=\sum_{i,k=1}^n P(A_iA_k),\quad U_n:=\sum_{1\le i<k\le n}P(A_iA_k).$$ Then $2T_n\le S_n^2\le2T_n+S_n$, and $S_n<<S_n^2 …

$\newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operato …

If $\xi$ is symmetric with $E|\xi|=1$, then $s=2\sup_{x>0}xp(x)$ is the supremum of the density $x(2p(x))/E|\xi|$ of the so-called size-biased distribution of $|X|$; see e.g. Arratia and Goldstein.

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

I have obtained a copy of the paper by Lakshmanamurti (1950); however, it appears that I cannot post a link to it here. Using Lagrange multipliers in a standard fashion, the problem is quickly reduced …

The order of magnitude of the sum of the first displayed series is indeed $\asymp n^2$ when the random walk is symmetric. Let \begin{equation} M_k:=\max_{0\le j\le k}X_j. \end{equation} Then, by th …

An approach to approximation of multiple sums by integrals, including summing possibly divergent multi-index series and sums over polytopes is presented in this paper. See also references there, espec …

$\newcommand{\de}{\delta}$This is to provide a detalization/formalization of fedja's answer. For $r\in(0,1]$ and real $x$, let \begin{equation*} f_{0,r}(x):=(x^2+r^2)^2,\quad g_{0,r}(x):=\frac x{x …

$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}$Let $$f(x):=\sum_{n\in\Z}2^{-n}K\Big(\frac{x-2^{-n}}{2^{-n-2}}\Big) =\sum_{n\in\Z}2^{-n}K\Big(\frac{x-2^{-n}}{2^{-n-2}}\Big)1(x\in2^{-n}[3/4,5/4]),$$ …

This inequality is known. Indeed, inequality (1.1) by Kwapień and Szulga states that $$\Big(\int_S\Big(\int_T|f(s,t)|^q\mu(dt)\Big)^{p/q}\nu(ds)\Big)^{1/p} \\ \le \Big(\int_T\Big(\int_S|f(s,t)|^p\nu( …

You want to maximize \begin{equation*} R(E):=\frac{\int_E\ m}{\int_E\ M} \end{equation*} over all admissible sets $E$, that is, over all Lebesgue-measurable sets $E$ such that $\int_E\ M>0$, where …

Let us show a bit more, that $F_p(s):=P(D^{(p)}\le s)$ is nondecreasing in $|p-1/2|$, for any real $s$. That is, take any $p$ and $p_1$ in $(0,1)$ such that $|p_1-1/2|>|p-1/2|$, and let, for brevity, …

Here is a simple proof. Since $f$ is nonnegative and decreasing, there is a limit $c:=\lim_{x\to\infty}f(x)\ge0$, and $f\ge c$. So, $\infty>\int_A^\infty f(x)e^{kx}\,dx \ge\int_A^\infty ce^{kx}\,dx=\i …

For $a\in(0,1)$ and any real $u,v\ge0$, $$(u+v)^a\le u^a+v^a, \tag{1}\label{1}$$ with the equality if and only if $u=0$ or $v=0$. Now, $$|x-y|^a=|(x-z)+(z-y)|^a\le(|x-z|+|z-y|)^a \\ \le|x-z|^a+|z-y|^ …