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Of course not. E.g., let $T$ be the set of all natural numbers, let $U$ be any random variable (r.v.), let the $Y(t)$'s be iid standard normal, and let $X(t):=\min(U,Y(t))$ for all $t$. Then the condi …

$\newcommand{\ep}{\varepsilon}\newcommand{\vpi}{\varphi}\newcommand{\de}{\delta}$Yes, this is true: By the reflection principle (see e.g. Proposition 2, for $M:=\max_{0\le t\le1}W_t$, \begin{equation} …

For general classes of bounded pdf's of $X_1$, including pdf's with exponential-like, super-exponential, and sub-exponential tails, your Theorem follows from the considerations in Sections 2.1 and 2.2 …

$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\la}{\lambda}\newcommand{\be}{\beta}\newcommand{\Y}[1]{\hat Y_{2n+1}^{#1}}$Since $\Y t$ is the sum of $2n+1$ independent copies of $Y: …

$\newcommand{\ep}{\varepsilon}$A bound similar to, and at least in some cases more accurate than, the bound presented in my previous answer on this page, can be obtained, a bit more easily, using form …

For each natural $i$, let $S_i$ and $T_i$ denote, respectively, the starting and terminal (ending) time moments of the $i$th stalling period. Let $D_i:=S_i-T_{i-1}$, the duration of the time between t …

Your inequality is trivial and useless as written. On its left-hand side we have a probability which is $\le1$ and goes to $0$ as $t\to\infty$, whereas on the right-hand side we have an expression whi …

Here it is more convenient to consider the order of magnitude of $S_n:=\sum_1^n X_i$, rather than that of $S_n/n$. Take any real $c>0$. Let $x:=cn^{1/p}$, $Y_i:=X_i\,1(X_i<x)$, $T_n:=\sum_1^n Y_i$, $M …

$\newcommand\Ga\Gamma\newcommand\Z{\mathbb Z}\DeclareMathOperator\Pois{Pois}$Let $t\mathrel{:=}\lambda$ and $k\mathrel{:=}x\in\Z\cap[0,t)$. Then for $X_t\sim \Pois(t)$ we have $$P(X_t\ge k)=1-\frac{\G …

Indeed, if the probability mass function of an integer-valued random variable is log concave as a function on $\mathbb Z$, then the corresponding cdf is also log concave as a function on $\mathbb Z$. …

$\newcommand{\ep}{\epsilon}$Somehow, I have only now recalled about Latala's inequalities for moments of the sums of positive independent random variables (r.v.'s), which, in particular, allow one to …

If (say) $s:=\sqrt{EX_1^2}<\infty$ then, by the central limit theorem, $Z_n:=S_n/n^{1/2}\to sZ\sim N(0,s^2)$ (as $n\to\infty$ in distribution). Also, $EZ_n^2=s^2<\infty$ and hence the sequence $(|Z_n| …

$\newcommand\ep\epsilon $Let $u:=\eta>0$, so that the probability in question is $P(\ln X>E\ln X+u)$. Note that this probability will not change if we replace there $X$ by $tX$ for any real $t>0$. So, …

Let $n:=N$. Let us show that for all natural $n$ and all $p\in(0,1)$ $$P(A_n>\sqrt p)\le\frac{\sqrt p+p}{1+p},\tag{1}$$ so that $P(A_n>\sqrt p)\to0$ whenever $p\downarrow0$. Consider first the case wh …

For $X_1:=X$, $X_2:=X'$, some positive real $c_1,c_2,a_1,a_2$, and all positive real $t$ we have $$P(|X_j|>t)\le c_j e^{-a_j t^2} \tag{1}$$ for $j=1,2$. So, for $t:=\epsilon>0$, $$P(|X_1-X_2|>t)\le P …