Questions tagged [probability-theory]

For questions solely 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 [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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If $B$ is a Brownian motion independent of $X$, then $B$ is still a Brownian motion under the regular conditional probability given $X$

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $B$ be a one-dimensional Brownian motion. Let $X, Y:\Omega\to \mathbb R$ be random variables. Let $\nu: \mathbb R \times \mathcal F \...
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Why do singleton events imply sets in multiplicative but not in additive probability?

Let $a \geq 0$ and $0\leq b \leq 1$ and $M, N$ be two appropriate conditioning events such that, for all singletons $y = \lbrace y \rbrace$ in the sample space $Z$ and all subsets $Y$ of $Z$, the ...
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Different measurability of Hilbert-space valued random variable

My question is motivated by this link. Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable. Now let $H$ be a ...
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0 votes
1 answer
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Using derivative as probability density function

Let be $X$ an absolutely continuous random variable. Then, we know that there exists a probability density function (pdf) $f_X$ such that its cumulative distribution function (cdf) $F_X(x)$ can be ...
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1 answer
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understand proof of strong law of large numbers (using reverse martingale)

I am reading the proof of strong law of large numbers using reverse martingale here. There are several points about conditional expectation that I do not understand. They all seem pretty intuitive but ...
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1 vote
1 answer
31 views

How can I show that $(\|B_t\|^2-dt)_{t\geq 0}$ is a martingale?

Let $B$ be a $d$-dimensional $(\Bbb{F}, \Bbb{P})$-Brownian motion where $\Bbb{F}=(\mathcal{F}_t)_{t\geq 0}$. Then consider $X:=(X_t)_{t\geq 0}=(\|B_t\|^2-dt)_{t\geq 0}$. I want to check that it is a ...
1 vote
1 answer
60 views

If $\mathbb EX=0$ and $\mathbb E|X| < \infty $ implies finite Variance?

I think, I am totally wrong, but if $\mathbb E|X|$ is finite then, $$\text{Var}|X|=\mathbb E|X^{2}|- (\mathbb E(X))^{2}=\mathbb E|X^{2}|< \infty$$ Considering $\mathbb E|X|< \infty$ implies $\...
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3 votes
2 answers
128 views

Probability and Random Variables.

Hi, I was trying to understand this example in the book. In the first part of the question, We've to find p.d.f (probability density function). For that, we take the derivative of the given ...
0 votes
0 answers
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`Cropped' $\max_{0\leq k\leq n}S_k - S_n$, where $(S_n)_{n\in \mathbb{N}}$ is a simple random walk, has the same distribution as $|S_n|$.

Let $S_n$ be a simple random walk with $S_0=0$ and $X_n:=\max_{0\leq k\leq n} S_k - S_n$ is a non-decreasing process which is zero when $n=T_n:=\max\{k\leq n: \sup_{0\leq k\leq n} S_k = S_n\}$. I want ...
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1 answer
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Can we characterize conditional independence by conditional probability?

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $(S, d)$ be a Polish space and $\mathcal S$ its Borel $\sigma$-algebra. Let $\mathcal A$ be a sub-$\sigma$-algebra of $\mathcal F$. ...
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1 answer
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Young's inequality in probability theory

The following is the standard version, in two equivalent statements: If $a \geq 0$ and $b \geq 0$ are nonnegative real numbers and if $p > 1$ and $q > 1$ are real numbers such that $\frac{1}{p} ...
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Conditional independence and conditional probability

I'm trying to make clear about the notion of conditional independence. Could you have a check on my below understanding? Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $\mathcal A, ...
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Interchanging expectation and integration with a collection of random integrands

Suppose one has a collection of i.i.d. random functions $\{f(\cdot,t):t\in\mathbb R\}$, where we write $f$ for the common distribution. In a probability application, what I need is to interchange ...
0 votes
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11 views

Integral with respect to the Brownian sheet/Wiener Field [duplicate]

Let a Brownian sheet/wiener field, $W\left(t_1, \ldots, t_d\right)$, for some $d \in \mathbb{N}$ ( see this : https://encyclopediaofmath.org/wiki/Wiener_field for a definition). For the $d=1$ case, we ...
1 vote
1 answer
49 views

Probabililty of sums of gaussians

Let $X\sim N(\mu_1, \sigma_1)$ and $Y\sim N(\mu_2, \sigma_2)$. Consider an arbitrary $r\in\mathbb{R}$. How can I compute the probability of $\mathbb{P}(X \leq r \leq X + Y)$? My idea: One could write ...
6 votes
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91 views

When does the power series $\sum_{n=1}^\infty a_nx^n$ have infinitely many zeros in $(0,1)$?

Let $f(x)=\sum_{n=1}^\infty a_nx^n$ be a power series, where $a_n=\pm 1$. I have encountered a probability question, saying that $(a_n)_n$ is a sequence of i.i.d. Bernoulli random variables, with $\...
0 votes
1 answer
66 views

On the Cauchy-Schwarz inequality: can there be something in between?

For any couple of random variables $X$ and $Y$, $$ \displaystyle |\mathbb{E} (XY)|^{2}\leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2}) $$ Is it possible to put an intermediate bound, like $$ \displaystyle |\...
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On the proof of Ito Isometry using partial sums

This post on MSE provides a nice derivation, but I don't understand this crucial step $$\begin{align}&\mathbb E\left[\sum_{i≠ j}K_iK_j(M_{t_{i+1}} - M_{t_i})(M_{t_{j+1}} - M_{t_j})\right] \\ =&...
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Mathematics Behind Coincidence of Probabilities (Magic the Gathering Example)

In Magic the Gathering, a player must decide how many land cards to include in his or her deck. Drawing too few at the beginning of a game is called mana screw. Drawing too many at the beginning of ...
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-1 votes
1 answer
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Mean square convergence of the squares

Let $X_n$ be a sequence of random variables converging in mean square to $X$, that is, \begin{equation}\tag{1} \lim_{n\to\infty}\mathbb{E}\Big(\big|X_n-X\big|^2\Big)=0 \end{equation} or, in short-hand ...
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1 vote
1 answer
28 views

An inequality about the 2-Wasserstein distance

Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
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1 vote
1 answer
48 views

If a sequence of random variables converges in probability to $0$, will the sequence still converge to $0$ when multiplied by $\sqrt n$

Assume we have a sequence of random variables $X_1,X_2...$ such that the sequence converges to $0$ in probability. Under what general conditions will the sequence still converge in probability to $0$ ...
0 votes
0 answers
28 views

Ito diffusion crosses initial value infinitely many times?

Consider the stochastic process $Y_t=y+\int_0^t\sigma(Y_s)dW_s$ where $W$ is a standard Brownian motion. It is well known that if $\sigma(y)=\sigma >0$ constant we have that $Y_t$ crosses $y$ ...
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0 votes
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16 views

Gaussian concentration to upper bound the variance of the norm of a gaussian vector

I want to prove the following: Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$. Then: $ Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2 $ This is ...
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1 vote
1 answer
42 views

Calculating Prize Line Expectation Part 2

Thanks in advance for any help. Yesterday a very helpful member called @joriki answered my original question on this and that conversation came to a conclusion as a result. I have a second part that ...
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1 vote
1 answer
35 views

Time interval between the first and the last maxima of random walk $\{S_k\}_{k=1}^n$ as $n\to\infty$

Let $S_n:=\sum_{i=1}^n X_i$ where $\{X_i\}$ are IID with $\mathbb{E}X_i = 0$ and $\mathbb{E}X_i^2=1$ for all $i\in \mathbb{N}$. We define stopping times: $\tau_n:=\inf\{1\leq k\leq n: S_n = \max_{1\...
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1 vote
0 answers
30 views

Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity

I am wondering about how to deal with a limit of the following form $$ \lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$ with the function $a_n(t)$ and the sum being dependent on $n$. Consider the ...
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1 vote
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52 views

Proof that given a continuous random variable $X$ with density function $f(x)$ then $\mathbb E[g(X)] = \int_{-\infty}^{\infty} f(x)g(x)dx$.

I have tried a million things and I just can't seem to crack this. To repeat it the question is, given a continuous random variable $X$, with density function $f(x)$ prove that: $$ \mathbb E[g(X)]=\...
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2 votes
1 answer
36 views

Expectation for an $x_i$ in a permutation.

Any tips on how to approach the next steps of this problem? $X$ is a set of $N ≥ 2$ distinct numbers and let $x_1x_2\cdots x_n$ be a permutation of $X$. For $i = 2, 3, . . . , n$ we say that position $...
1 vote
1 answer
38 views

MLE for the sum of a normally distributed variable and constant after a specific time [closed]

We start off with a normally distributed random variable $X$ with known $\mu=100$ and $\sigma^2=1$, and after $\vartheta$ days, a constant $1$ gets added to the value each day. Given $X_1,...,X_n$ ...
0 votes
1 answer
33 views

In what way is a Gaussian process a distribution over function space?

Gaussian processes are generally introduced as families of rvs where all finite vectors are multivariate normal. However, they are also described sometimes as "distributions over functions." ...
-2 votes
1 answer
59 views

The two random variables $X$ and $Y$ have the following common distribution

EDIT: here is the formatting of the problem. I already solved a) and b), I attempted to solve c) and struggle with d) $$ \begin{array}{c|lcr} X \setminus Y & 0 & 1 \\ \hline -1 & 0 & ...
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1 vote
1 answer
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Let $M$ be a continuous martingale, $r >0$, and $\tau := \inf \{t \ge 0 : \langle M \rangle_t \ge r \}$. Then $M_\tau$ is square-integrable

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous martingale w.r.t. $\mathcal G$ such that $$ (\...
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2 votes
1 answer
48 views

Show that $T$ is a stopping time

Show that $$T = \inf \left \{n \geq 0|X_n \in \left \{0, N \right \} \right \}$$ is a stopping time with respect to $\mathcal{F}_n=\sigma(X_0,...,X_n)$ for $n\geq 0$. I am fairly new to stopping times....
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2 votes
2 answers
36 views

Given this weak property is it possible to demonstrate that the difference of expected value is negative?

Let's assume that we have $X,Y$ as random variables and we have as hypothesis that $$X-\mathbb{E}_x \leq Y-\mathbb{E}_y$$ where $\mathbb{E}_x$ is the expected value of x. Is it possible to demonstrate ...
0 votes
1 answer
28 views

Convergence of ratio of Big O-terms in probability

Say we have constants $v \in \mathbb{R}$ and $w >0$. Assume that $a_n \to 0$ and $b_n \to 0$ as $n \to \infty$. I wish to show that $$ \frac{v+O_p(a_n)}{w+O_p(b_n)}=\frac{v}{w}+O_p(a_n) $$ where we ...
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2 votes
2 answers
33 views

Let $X_n \to X$ and $Y_n \to Y$ in probability such that $B \subset \{X_n = Y_n\}$ for all $n$. Is $B \setminus \{X=Y\}$ a null set?

Let $(\Omega, \mathcal F, \mathbb P)$ be a complete probability space. Let $X, Y, X_n, Y_n:\Omega \to \mathbb R$ be random variables such that $X_n \to X$ in probability and $Y_n \to Y$ in probability....
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2 votes
1 answer
44 views

Calculating Expected Increase In Prize Line Count

Hello and thanks in advance. I'm trying to solve a problem I'm having with a pattern matching game. The game's board consists of a 5x5 grid of numbers where column 1 (on the far left) contains 5 ...
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0 votes
1 answer
25 views

Intuition of conditioning on future observation in brownian motion?

I've been learning about Brownian motion and have verified a result that for a Brownian motion process $\{B_t : t \geq 0\}$, $B_0 =0 $ and $0 \leq s < t$, $$ B_s | B_t \sim \mathcal{N}\left(\frac{s}...
-1 votes
0 answers
25 views

Finding the CDF and the PDF of random variable $ Y=C\sqrt{r^2−B^2}^D$

For a random variable given as $R$ with pdf $$f_R(r) = \frac{12}{73} \left(\frac{27r}{M^2}-\frac{35r^3}{M^4}+\frac{8r^5}{M^6}\right)$$ I need to find pdf and cdf $$ Y=C(\sqrt{r^2−B^2})^D$$ where $C , ...
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0 answers
52 views

How to fill in this proof of Jensen's inequality for conditional expectation?

Recall Jensen's inequality for conditional expectation: $X$ an integrable (or nonnegative), real-valued, $\mathfrak{F}$-measurable random variable. $\varphi: \mathbb{R} \to \mathbb{R}$ convex. $ \...
0 votes
0 answers
22 views

$\mathfrak{F}$ contains an even number of subsets of $\Omega$

Show that, if $\Omega$ is a finite set, then $\mathfrak{F}$ contains an even number of subsets of $\Omega$. Proof: Let $f:\mathfrak{F}\rightarrow\mathfrak{F}$ such that $f(A)=\Omega\setminus A,\text{ }...
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1 vote
1 answer
49 views

How to find a probability given other probability functions?

I have the following problem: consider 4 discrete RVs $a,b,c,d$ with $a, b$ independent. The following pmfs are known: $p(a)$, $p(b)$, $p(c|a,b,d)$, $p(d|a,b,c)$ for every value of $a,b,c,d$. Find $p(...
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0 votes
1 answer
37 views

Markov chains - train ride

Problem: A country has $m+1$ cities $(m\in\mathbb{N})$, one os which is the capital. There is a direct railway connection between each city and the capital, but no tracks between any two 'non-capital' ...
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1 vote
1 answer
61 views

How can I prove this equivalent relation about measurable random variables?

How can I prove “ random variable $\alpha$ is $\mathcal{F}\vee \sigma(E_k,k=0,1,…,N)$-measurable” if and only if “$\displaystyle{\alpha=\sum_{k=0}^N} \alpha_k\mathbf{1}_{E_k},\ $ where for any k, $\...
3 votes
1 answer
74 views

Distribution of pre-transformed Gaussian Random variables

Let $f,g:\mathbb{R} \to \mathbb{R}$ be monotonic and differentiable functions, and suppose that $(X, Y)$ is a random vector such that $$ (f(X),g(Y))^T \sim N (\mu, \Sigma), $$ where $\mu = (\mu_1, \...
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0 votes
1 answer
31 views

Partial order relation on $\mathcal{M}_{1,c}(\mathbb{R})$.

Let $\mathcal{M}_{1,c}(\mathbb{R})$ the space of Borel probability measures on $\mathbb{R}$ with compact support. There is a partial order relation on $\mathcal{M}_{1,c}(\mathbb{R})$: $$\mu\succeq \nu ...
0 votes
0 answers
31 views

On the autocovariance function of progressively measurable processes

Suppose I have a progressively measurable process $G_\omega(t)$ (adapted process + càdlàg paths), which satisfies $$\tag{1} \mathbb{E}_\omega\left(\int_{t_0}^t(G_{\omega}(t'))^2\mathrm{d}t'\right)<+...
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4 votes
0 answers
76 views

Convergence in probability and differentiability of characteristic function

I am trying to prove the following: If $(X_1+...+X_n)/n \xrightarrow{n \to \infty} m < \infty$ in probability then the characteristic function $\phi$ is differentiable in zero and $\phi'(0)=im$, ...
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0 votes
1 answer
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Differentiability in zero of characteristic function and expected value of i.i.i random variables

I would like to prove the following implication: If the characteristic function $\phi$ is differentiable in zero and $X_1\geq0$ a.s., then $E(X_1)=i\phi'(0)<\infty$. (This is Exercise 15.4.4 iii) ...
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