Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [almost-everywhere]

For questions about the concept of "almost everywhere", that is, questions about properties which holds everywhere, except on a set of measure 0. This is involved in probability theory as well as in the case of infinite measure space. Use the appropriated tags to specify the context.

2
votes
1answer
37 views

Showing that for $\{u_n,u\}_{n \geq 1} \subseteq L^p(\Omega)$ it is $u_n \to u$ in $L^p(\Omega)$.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $\{u_n, u\}_{n \geq 1} \subseteq L^p(\Omega)$ with $1<p<\infty$ and we assume that $\|u_n\|_p \to \|u\|_p, \; u_n \...
0
votes
1answer
50 views

Why finite integral implies convergence almost everywhere?

In this proof why finite integral implies convergence almost everywhere? Note there sum symbols $\sum$ are missing in the complete proof; not sure why Why having $\int g^p<\infty$ implies the ...
-1
votes
2answers
55 views

$N(t)/t\to 1/\mu$ [closed]

Let $X_i$‘s are iid with $0<X_i<\infty$ almost surely. Define $T_n=X_1+\cdots+X_n$ E$(X_1)>0$ N$(t)=\sup[n\ge 0: T_n\le t]$ How to show using Strong law of Large Numbers that $N(t)/t\to1/\...
1
vote
1answer
50 views

Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
2
votes
1answer
59 views

A criterion of convergence almost surely.

Suppose random variables $\{X_k\}_{k\in \Bbb{N}}$ are $i.i.d.$ and set $S_n=X_1+...+X_n$, show that if $S_n/n\rightarrow 0$ in probability and $$S_{2^n}/2^n\rightarrow 0 \ \ a.s.$$ then $S_n/n\...
0
votes
0answers
28 views

$X_n$ converges in probability but not almost surely

Consider the sequence $\{X_n\}$ given in Davide Giraudo's answer to this question. As is explained in the answer, the lack of a.s. convergence comes from the Borel-Cantelli lemma. Another way of ...
1
vote
0answers
25 views

Does the specific sequence of random variables converge almost surely to a given constant?

Suppose, $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d random variables, such that $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$. Now, suppose $\{S_n\}_{n = 1}^\infty$ is a sequence of random variables ...
2
votes
2answers
65 views

Problem with left limits.

Let $F: \mathbb{R} \to \mathbb{R}$ a non-decreasing function and suppose $G: \mathbb{R} \to \mathbb{R}$ defined by $G(x) = F(x+)$ (= the right limit of $F$ in $x$, which always exists for non-...
1
vote
2answers
49 views

A random variable $X$ is $P$-a.s. constant iff $\sigma(X)$ is $P$-trivial - for which state spaces?

Let $(\Omega,\mathcal F, P)$ a probability space, $(E,\mathcal E)$ a measurable space, and $X\colon\Omega\to E$ a ($\mathcal F$-$\mathcal E$-measurable) random variable. In the case $(E,\mathcal E)=(\...
0
votes
1answer
25 views

Is this sequence almost sure convergent?

Consider the sequence of independent random variables $\{X_n\}$ such that $$\begin{align} P(X_n = 1) &= 1/n \\P(X_n = 0) &= 1 - 1/n \end{align}$$ I saw this as an example of convergent in ...
0
votes
1answer
46 views

Modification of the law of large numbers for Binomial random variables. [duplicate]

Let $(p_n)_{n \geq 1}$ be a sequence of numbers in $[0,1]$ such that $p_n \to p$. Let $(X_n)_{n \geq 1}$ be a sequence of independent random variables where $X_n \sim Bin(n,p_n)$. Is it true that $X_n/...
0
votes
1answer
38 views

Does $P(\liminf_{n \to \infty}\{|X_{n}|\leq \epsilon\})=1\iff \exists N \in \mathbb N, |X_{n}|\leq \epsilon, \forall n \geq N, P-$a.s.

Background to my question: Given that $(X_{n})_{n}$ are random variables on $(\Omega, \mathcal{F}, P)$ and for $\epsilon > 0$: $\sum_{n \in \mathbb N}P(|X_{n}|>\epsilon)<\infty$ It follows ...
1
vote
1answer
29 views

Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\infty$ almost surely

Let $(X_{n})_{n}$ be a sequence of random variables that are identically distributed on $\mathcal{U}(0,1)$. Furthermore, let $\alpha > 1$. Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\...
1
vote
0answers
33 views

How to show $\int_{0}^{1}|f|^{p}|g_{n}-g|^pd\mu$

Let $(g_{n})_{n}$ be bounded and measurable where $g_{n}\xrightarrow{n \to \infty} g, \mu-$a.e. and $f_{n} \xrightarrow{L^{p}}f$. I need to show that $g_{n}f_{n} \xrightarrow{L^{1}}gf$, and my proof ...
1
vote
1answer
25 views

Proof of converge in probability

For $\epsilon>0$ define $E_{n}=\begin{Bmatrix}\omega:|X_{n}(\omega)-X(\omega)|>\epsilon\end{Bmatrix}$, If $X_{n}\overset{a.e}{\rightarrow}X$, then show that $\mathbb{P}(\cup_{n\geq m}E_{n})\...
3
votes
0answers
88 views

Problem seems too easy. Is there a trap? Show that $f(1-f)=0$ almost everywhere…

So this is a problem I found on a qualifying exam and it has been nagging at me for some a while. It seems overly easy. Assume that $f$ is a nonnegative function on $[0,1]$ and \begin{align} \int_{[...
0
votes
1answer
38 views

Question regarding measurable set, Hausdorff-Space and almost everywhere properties of measurable functions f,g

I've been given the following task Let $(X,\mathscr{M}_X,\mu)$ a measure space. Two measurable mappings $f,g:X \to Y$ into a measurable space $(Y,\mathscr{M}_Y)$ are called equal almost ...
2
votes
1answer
42 views

Show that this is an inner product

Let's define $$(f,g)=\int_{\mathbb{R}} \frac{f(x)\bar{g}(x)}{1+x^2}dx$$ $\forall f,g\in X=\{h:\mathbb{R}\rightarrow\mathbb{C}:$ $h$ is Lebesgue-measurable and bounded over $\mathbb{R}$} I have to ...
1
vote
0answers
67 views

If $P(\max_{n \leq i \leq j \leq k \leq m} \min \{|S_i - S_j|, |S_k - S_j|\} \geq \lambda)$ is bounded appropriately, does $\sum \xi_j$ converge a.s.?

Suppose that $$E\left[|S_j - S_i|^\gamma |S_k - S_j|^\gamma \right] \leq \left(\sum_{i < l \leq j} u_l\right)^\alpha \left(\sum_{j < l \leq k} u_l\right)^\alpha$$ for $0 \leq i \leq j \leq k \...
1
vote
0answers
50 views

Brezis 4.15, $f_n(x)=ne^{−nx}$

Let $\Omega=(0,1)$ and $f_n(x)=ne^{-nx}$ prove that (i) $f_n\rightarrow0$ a.e. (ii) $f_n$ is bounded in $L_1(\Omega)$ I know that for $f_n$ to converge a.e. to zero, the set of points in which this ...
0
votes
0answers
37 views

For which x the following sequences converges

If $q_n$ be an enumeration of rational numbers, for which $x$ the following sequence converges? $$\sum_{n=1}^{\infty}e^{-n^2|x-q_n|}.$$ I guess that for no $x$ the sequence converges. I tried to ...
1
vote
3answers
34 views

Almost sure convergence of a martingale sum

Consider the senquence of iid r.v. $(Y_k)_{k\geq1}$ such that $\mathbb{P}(Y_k=1)=\mathbb{P}(Y_k=-1)=\frac{1}{2}$ and then consider the process $X=(X_k)_{n\geq1}$ such that $X_n=\sum_{k=1}^n\frac{Y_k}{...
1
vote
1answer
35 views

Function almost always periodic for every period is constant

Let be $(X, \mathcal{A}, \mu)$ a measure space and $f: X \to [0, +\infty]$ a measurable function such that for every $x \in X$ we have $f(t + x) = f(t)$ for almost every $t \in X$. Is true that $f$ is ...
0
votes
1answer
37 views

If $E[|\sum_{i < l \leq j} \xi_l |^\gamma] \leq (\sum_{i < l \leq j} u_l)^\alpha$ and $\sum u_l < \infty$ then $\sum \xi_l$ converges almost surely

Suppose $\xi_1, \xi_2, \ldots$ are random variables that satisfy $$E\left[\left|\sum_{i < l \leq j} \xi_l \right|^\gamma\right] \leq \left(\sum_{i < l \leq j} u_l\right)^\alpha$$ for $\gamma \...
4
votes
0answers
104 views

$\int_A f = \int_A g \implies f = g$ a.e.

Consider a measure space $(\Omega, \mathcal{F}, \mu)$ and let $f,g: \Omega \to \mathbb{R}$ be $\mathcal{F}$-measurable integrable functions. If $$\int_A f d \mu = \int_A g d \mu$$ for all $A \in \...
0
votes
1answer
38 views

Function in $L^2$ that doesn't vanishing

Is this statement makes a sense: $f\in L^2(0,1)$ such that $f(x)\ne 0 ,\forall x\in (0,1)$ ?
1
vote
2answers
63 views

$f$ is finite a.e if it is locally integrable?

I know that when a function $f$ is integrable then it is finite almost every where. I was wondering to know if it is true that whenever $f$ is locally integrable then $f$ is finite almost everywhere. ...
1
vote
2answers
75 views

“Almost uniformly convergent” Implies “Uniformly convergent almost everywhere” . Is there something wrong?

I find a tricky proof shows "almost uniformly convergent" implies "uniformly convergent almost everywhere". I know it is wrong, for there is a counterexample. Can anyone help me why this proof is ...
1
vote
1answer
30 views

Manipulations with convergence a.e.

Let functions $f_n$ be measurable, $n \in N$, $f_n\rightarrow f$ almost everywhere. Prove that $\operatorname{arctg}f_n \rightarrow \operatorname{arctg}f$ almost everywhere. Honestly speaking, we ...
2
votes
3answers
53 views

$\int_Xfgd\mu=0$ $\forall g$ in a dense subset of $L^q$ => f=0 a.e.

Let $(X,\mathscr{E},\mu)$ be a measure space, $p\in(1,\infty)$, $q=\frac{p}{p-1}$ and $S\subset L^q$ be a dense subset. Then $$f\in L^p \text{ and } \int_Xfgd\mu=0 \text{ } \forall g\in S => ...
0
votes
1answer
103 views

Prove that almost surely $\lim\sup_{n\to\infty}\frac{X_n}{\ln n}=\frac{1}{\lambda}$

Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(\lambda)$. Prove that almost surely $$\lim\sup_{n\to\infty}\frac{X_n}{\ln n}=\frac{1}{\lambda}$$ My idea was to look at ...
1
vote
1answer
34 views

$\lambda$-almost-everywhere convergence implied by lim$\lambda (${$x \in E : | f_n(x) - f(x) | > \epsilon$}$) = 0$

Let $E \subset \mathbb{R}$ be $\lambda$-measurable and let $f_n,f: E -> \mathbb{R}$ $\lambda$-integrable, so that for all $\epsilon > 0$ $\lambda(${$x\in E: |f_n(x)-f(x)| > \epsilon$}$)->...
1
vote
0answers
26 views

Prove the upper bound of expectations using almost sure convergence

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of iid random variables. Given that $X_n$ converge almost surely to $x_0\in [m, M]$, where $0<m<M<1$, I try to find the upper bounds of $\mathbb{E}\...
0
votes
1answer
31 views

If either $f_n \longrightarrow f $ almost everywhere OR $f_n \longrightarrow f$

[1] Are all simple functions Lebesgue Integrable? Prove or disprove. [2] If either $f_n \longrightarrow f $ almost everywhere OR $f_n \longrightarrow f$ in measure then show that f is finite valued ...
0
votes
0answers
67 views

If $f_n \longrightarrow f$ almost everywhere and $g_n \longrightarrow g$ almost everywhere

1 The Riemann function $f:[-1,1]\longrightarrow \mathbb{R}$ defined as $$ f(x)= \begin{cases} 0, &\textrm{if }x \notin \mathbb{Q}\cap[-1,1]\textrm{ or }x =0\\ \frac{1}{q} , &\textrm{if }x\in ...
2
votes
1answer
102 views

$f = g$ almost everywhere implies $f =g$ for any continuous function $f$ and $g$.

Let $f,g : [0,T] \to X$ be continuous $X$-valued functions for Banach Space $(X,||\, \cdot\,||)$. Suppose that $f = g$ almost everywhere in $C([0,T],X)$, then $f = g$ in $C([0,T],X)$. This is my ...
1
vote
1answer
31 views

Relationship conditional expectation and random variable under specific constraint on its values

I am trying to establish a relationship between the following conditional expectation and random variable based on the a given identity: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
2
votes
1answer
59 views

Prove that ${S_n/n}$ does not converge a.s.

This is an old qualifying exam question of probability theory. Let $\{X_n\}$ be a sequence of independent random variables with $X_1=0$ and for $k\geq 2$, define $X_k$ as $$\mathbf{P}(X_k=k)=\mathbf{...
0
votes
0answers
54 views

Cardinal of set goes to infinity a.s.

Supposem $X1,X2,...$ are i.i.d. and $X1$ takes each value $j$ with probability $p_j$. Let $D_n$ be the number of distinct values $j$ among $X1,X2,...X_n$, that is, $D_n=|{X1,X2,...,X_n}| (i) Show ...
3
votes
1answer
83 views

Weak convergence of measures implying almost sure convergence of random variables

Suppose $\mu,\mu_n$ are Borel probability measures on $\mathbb{R}$ with $\mu_n$ converging weakly to $\mu$. I am asked to find some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and random ...
1
vote
0answers
58 views

Is a sequence diverging almost surely to infinity almost surely positive?

I have proved that a sequence of random variables $(M_n)_{n\in\mathbb N}$ diverges to $+\infty$ almost surely. I.e I have proved that $$\bigcap_{c\in\mathbb Q^+}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^\...
3
votes
1answer
59 views

Almost everywhere measurable function definition

Given $(\Omega, \mathcal{M})$ measurable space and $(X,\tau)$ topological space. A function $f: (\Omega, \mathcal{M}) \to (X,\tau) \ $ is almost everywhere measurable if $ \exists \ \Omega_0 \...
1
vote
1answer
79 views

Product of Uniform Distribution

I know that there exists some discussions related to my question, however, I couldn't find an explanation for my question. I hope it is not a duplicate. Let $X_n$ be sequence of i.i.d. uniform ...
0
votes
2answers
70 views

Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?

Let $\{X_n\}_{n=0}^\infty$ be an absorbing Markov chain. It is well-known that $$ P(\text{chain gets absorbed}|X_0=i)=1. $$ My question is how this is interpreted in practice. We have that for almost ...
2
votes
1answer
47 views

Probability path - Exercise 6.14 : on almost sure divergence

Let $\{X_n\}$ be independent with $P(X_n = n^2) = \frac{1}{n}$ and $P(X_n = -1) = 1 - \frac{1}{n}$. Show that $\sum_{n=1}^{\infty}X_n = -\infty$ almost surely. I found that $E[X_n] = n + \frac{...
0
votes
1answer
74 views

Generalized Egoroff's Theorem

Let $(X, \mathcal{A}, \mu)$ be a measure space. If $(f_n)_{n \geq 1}$ is a sequence of functions such that $|f_n| \leq g$, for such $g \in L^1(X, \mu)$, $f_n: X \rightarrow \mathbb{R}$, $f: X \...
1
vote
1answer
94 views

Simple example illustrating “almost sure convergence”

I'm trying to check my knowledge on th enotion of almost sure convergence. I've cooked up this example: Let $X_1$ be a Bernoulli random variable with paramter $\frac{1}{2}$. Now consider the sequence ...
1
vote
0answers
39 views

Checking almost sure converge of a random variable experimentally

Suppose $X_1, X_2, X_3, \ldots$ is a sequence of random variables that converges almost surely to a random variable $X$. How could I check for this experimentally? I don't need this to be some ...
2
votes
1answer
103 views

Proving almost sure convergence (help understanding a step in a proof)

We have $$|V_{n+1}-V^\prime_{n+1}|=\prod_{m=1}^n|W_m||V_1-V^\prime_1| \tag{1}\label{1}$$ where $P(|W|=1)\ne 1$ (with $W\in[-1,1]$) and $W_n$s are iid and independent of $V$s and $V^\prime$s. Show ...
1
vote
1answer
50 views

Does $\operatorname EX_n\to0$ as $n\to\infty$?

Suppose that $X_1,X_2,\ldots$ are non-negative random variables defined on a probability space $(\Omega,\mathcal F,P)$ with $\operatorname E|X_n|^p<\infty$ with some $p>2$ for each $n\ge1$. ...