Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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Proving almost sure convergence

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ ...
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61 views

$X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite ...
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1answer
99 views

Probability of a set that has infinite Lebesgue measure

Forgive, for the title didn't know how to name this questions. Please change to something better. Let $B_1(n)$ denote a unit ball around $n\in \mathbb{Z}^{+}$. Suppose that for every $n$ there exists ...
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70 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuous stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
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80 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
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1answer
470 views

Conditional expectation w.r.t. random variable and w.r.t. $\sigma$-algebra, equivalence

Let $\Omega = \{ \omega_i \}$ be a countable set, and consider some probability space $(\omega, \mathcal F, P)$ with $p_i := P(\{ w_i \})$. Let $X : \Omega \to \mathbb R$ be a random variable, then ...
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1answer
468 views

Expectation of a discrete random variable: how to convert an integral to a sum?

According to wikipedia and all my textbooks, we define the expectation of a random variable on a probability space $(\Omega, \mathcal{F},P)$ : \begin{align} E(X) &= \int_{\Omega}XdP\\ \end{align} ...
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2answers
45 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
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573 views

Conditional expectation of indicator function

Could someone confirm if the following is correct. If not why? \begin{equation} E[\mathbb{1_{X\leq x}}|Y]=P[X|Y]=\frac{P[X,Y]}{P[Y]} \end{equation} Thank you.
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Show that $\sum_{n=1}^{\infty}\mathbb{P}\left(\frac{1}{n}\lvert X_n\rvert>\varepsilon\right)<\infty$.

Let $(X_n)$ be a sequence of identically distributed integrable random variables. My aim is to show for an arbitrary $\varepsilon>0$, that $$ ...
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1answer
177 views

Example of a sequence of r.v.'s with constant stopping time that is not a Martingale

Could anybody give me a simple example of a sequence of random variables $(X_{n})_{n \geq 0}$ that has constant expectation, but is not a martingale?
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102 views

Independence (probability theory)

Having hard time with this. Hope someone can help me! We throw a dice 20 times. $A(i,j)$ is the event in which in $i$-th and $j$-th throw we get same number. Show that $\{ A(i,j) :1\leqslant i\lt j ...
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236 views

When can we exchange expectation and maximum for asymptotic results?

Motivated in the analysis of algorithms, consider the following setup. Assume we have discrete random variables $X^{(n)}_1, \dots, X^{(n)}_n$ which we can not assume to be identical or independent. ...
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39 views

Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
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134 views

Deriving the transformation function of a random variable from the original and the final distributions

Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this ...
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3answers
234 views

Conditional expectation in the case of $\mathcal{A}=\{\emptyset,\Omega,A,A^c\}$

I have a small computation to do and I am not able to prove it: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $X$ be an integrable random variable and $A\in\mathcal{F}$. Let ...
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154 views

How to find the odds of a horse wining a race?

The odds against a horse winning are 2 to 1. What is the probability of this horse winning?
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214 views

Is it possible to derive the CDF of $Z$?

Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} ...
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1answer
2k views

Questions about independence between random variable and $\sigma$-algebra

Given a probability measure space $(\Omega, \mathcal{F}, P)$, if a random variable X and a sub $\sigma$-algebra $\mathcal{A}$ are independent, I was wondering why: $$E (X|\mathcal{A}) = (EX)I_Ω;$$ ...
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Sigma algebra proof problem

Let $\mathcal{F}$ be a $\sigma$-algebra of subsets of $\Omega$ and $B \in \mathcal{F}$. Show that $E = \{A \cap B: A \in \mathcal{F}\}$ is a $\sigma$-algebra of subsets of $B$. Is it still true when ...
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Will this procedure generate random points uniformly distributed within a given circle? Proof?

Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that ...
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1answer
102 views

If $A$ is the generator of $(P_t)$, then $A+f$ is the generator of $(P_t^f)$

Let $X=(X_t)_{t\geq0}$ be a Markov process on a state space $\Gamma$ (a Hausdorff topological vector space), let $A$ be the infinitesimal generator of $X$ and let $\mathcal C(\Gamma)$ the space of ...
7
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279 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
6
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1answer
161 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
6
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1answer
278 views

Squeeze theorem for convergence in distribution

Let $(X_n)_{n\in\mathbb{N}}$, $(X'_n)_{n\in\mathbb{N}}$, $Y$ real random variables such that $$ X_n\leq Y\leq X'_n \quad\forall n\in\mathbb{N}\ .$$ Suppose that the sequences $X_n$ e $X'_n$ converge ...
6
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2answers
632 views

coupon collector and Markov chains

I need some help with my homework in probability. I need to prove that if $X(n) =$ the number of different coupons that the collector has in time $n$ then $X(n)$ represents a Markov chain. I proved ...
6
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2answers
390 views

Confidence band for Brownian Motion with uniformly distributed hitting position

Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that $$ ...
6
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1answer
921 views

What is the expected number of runs of same color in a standard deck of cards?

Standard deck has $52$ cards, $26$ Red and $26$ Black. A run is a maximum contiguous block of cards, which has the same color. Eg. $(R,B,R,B,...,R,B)$ has $52$ runs. $(R,R,R,...,R,B,B,B,...,B)$ has ...
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171 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
5
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1answer
271 views

Intuition behind measurable random variables and $\sigma$-algebra

I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture. I'll try to ...
5
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3answers
207 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
5
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1answer
68 views

How to show that $p$th moment being finite is equivalent to a limit existing

Let $p \in (0,2)$ and let $\xi_n, n \geq 1$, be iid random variables. Show that the following two conditions are equivalent: With probability one, the limit $$ \lim_{n \rightarrow \infty} ...
5
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1answer
208 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
5
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1answer
169 views

Probability convergence in distribution

$Y_1, Y_2,..., Y_n$ are i.i.d and uniformly distributed on the set $\{1, 2,..., n\}$. Define $X_n = \min\{k: Y_k = Y_j\; \text {for some}\; j < k\}$, and prove that $\frac {X_n}{\sqrt n}$ converges ...
5
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2answers
707 views

Is $\{\sin(\omega n), n \geq 1\}$ a strictly stationary process?

Let $X(t)=\sin(\omega t)$, where $\omega$ is is uniformly distributed R.V. on $[0,2π]$. Let $X_n=X(n)$, is $\{X_n,n \geq 1\}$ a strictly stationary process? I've calculated that the distribution ...
5
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1answer
179 views

An Integral Involving Brownian Motion

Let $B_t$ $(t \geq 0)$ be a Brownian motion on $\mathbb{R}^3$. That is, $B_t = (B_{t}^{(1)},B_{t}^{(2)},B_{t}^{(3)})$, where each $B_{t}^{(i)}$ is a Brownian motion on $\mathbb{R}$. Let $Y$ be a Borel ...
5
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3answers
9k views

Repeating something with (1/n)th chance of success n times

Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every ...
5
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1answer
2k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
5
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2answers
2k views

Weak convergence of probability measure

I am working on a problem. Show that for each probability measure $\mu$, there exists probability measure $\mu_n$ with finite support such that $\mu_n$ converges weakly to $\mu$. I am thinking about ...
5
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1answer
599 views

Is Lindeberg's condition satisfied when variances of the sequence of r.v.'s are bounded from above and below?

This question is related to my previous question but I think it's sufficiently different to warrant a separate one. Suppose I have a sequence of independently distributed positive random variables: ...
5
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1answer
906 views

How does one prove that Lindeberg condition is satisfied?

I have a sequence of $n$ random variables, each drawn from a different distribution $X_1\sim A_1, X_2\sim A_2, \ldots, X_n\sim A_n$. The distributions $A_1, A_2, \ldots, A_n$ have nice properties: ...
5
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1answer
191 views

Existence of a completely supported probability measure

Given a compact Hausdorff space $X$, does there exist a probability $\mu$ on X such that the support of $\mu$ is $X$? This is equivalent to say, for any unital commutative C*-algebra, can we show the ...
5
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1answer
1k views

Inner product and norms for random vectors

From wikipedia inner product page: the expected value of product of two random variables is an inner product $\langle X,Y \rangle = \operatorname{E}(X Y)$. How it can be generalized in case of random ...
5
votes
1answer
871 views

Upper bound on the l1 norm of a multivariate normal random variable

Let $X \sim {\cal N}_d(0, \sigma^2I_d)$. I am interested in bounding the tail probability $P[||X||_1 > t]$ from above. A pointer to a known exponential or polynomial tail bound would be ...
4
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1answer
91 views

Question on part 3 of the Star Trek problem in Williams, Probability with Martingales

Consider this M.SE question, which is E12.3 in Williams. The answer of Robert Israel (and Xoff) seems to give an exponential bound on $R_n$ almost surely. Wouldn't this imply the convergence of ...
4
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1answer
47 views

Stopping time in Markov chains

A random variable $T : \Omega \rightarrow ${$1,2,3...$} $\cup$ {$ \infty$} is called a stopping time if the event {$T=n$} depends only on $X_0 , X_1 ,X_2 ,..., X_n$ for $n = 0,1,2,...$ I have trouble ...
4
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1answer
57 views

Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$? If the result is not true for ...
4
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1answer
115 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
4
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2answers
176 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
4
votes
2answers
217 views

Sample path of Brownian Motion within epsilon distance of continuous function

Given a continuous function $f:[0,1]\rightarrow\mathbb{R}$, $f(0)=0$, how can one show that $P(\underset{0\leq t\leq1}{\sup}\left|B_{t}-f(t)\right|<\varepsilon)>0$, where $P$ is the probability ...