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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
4 views

Notation in Kuo's Introduction to Stochastic Integration

I am reading this book by Kuo (really like it so far) and I really don't understand the notation used in section 10.5. In particular, when he writes $(X_{t_2} \in dx_2)$. I can't find anywhere else... ...
user avatar
  • 88
0 votes
0 answers
13 views

Variance for random variable with known density

let $t,h,g\in \mathbb{R}$ and $$h(z|x)=\frac{1}{\sqrt{2\pi}}e^{-1/2(z-t-hx-gx^2)^2}$$ $$f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ and $g(x,z)=f(x)h(z|x)$. Find the variance of $Z$ I have found the ...
user avatar
-1 votes
0 answers
24 views

If $X$ and $Y$ are independent $\implies$ $E(X|Y)=E(X)$

Exercices : Let $X$ et $Y$ be a random variables integrable such that $XY$ also integrable . Prove that : if $X$ and $Y$ are independent $\implies$ $E(X|Y)=E(X)$ $\implies~~~E(XY)=E(X)E(Y)$ My ...
user avatar
  • 2,135
0 votes
0 answers
12 views

Justify the differentiation in Durrett's investment problem.

I'm trying to solve the Exercise 2.4.3.(ii) of the Durrett's book, 5th edition. Here, $V_{i}\geq0$ is a random variable over the measure space $\Omega$ with $EV_{i}^{2}<\infty$ and $EV_{i}^{-2}<\...
user avatar
  • 1,595
0 votes
1 answer
13 views

Strong law of large numbers for p.w. uncorrelated integrable random variables with $M:=sup_{k}Var(X_k)<\infty $

A sequence of real integrable ($L_1(\Omega)$) random variables $(X_n)_{n\geq 1}$ is said to satisfy the $\quad$ 1) weak $\quad$ 2) strong $\quad$ law of large numbers in case $\overline{X_n}:= \frac{...
user avatar
  • 29
0 votes
0 answers
14 views

uniformly bounded Tail probabilities by the tail probability of another random variable

Consider the following setting. Let $1 < p < 2$, let $(X_i)_{i \in \mathbb{N}}$ be a sequence of non-negative (real-valued) random variables and let $(D_i)_{i \in \mathbb{N}}$ be a sequence of ...
user avatar
0 votes
0 answers
7 views

Ascending ladder variables and epochs when the time is constraint

Let $\{S_k\}_{k\ge0}$ be a random walk with $S_0=0$. Set $T_0=0$ and define $$ T_1=\inf\{k>0: S_k>0\} $$ and $$ T_j=\inf\{k>T_{j-1}:S_k>S_{T_{j-1}}\}, j\ge2. $$ If no such $k$ exists we ...
user avatar
  • 285
0 votes
2 answers
40 views

What is the probability of finding 2 in set of primes?

Let $$S=\{p :p\in\mathbb{P}\}$$ Be the set of primes. So $2$ is also a member of it right? $$P[\text{ finding 2 in $S$}]=\frac{1}{\infty}=0 ??$$ Does that mean that probability of finding $2$ from the ...
user avatar
  • 437
0 votes
2 answers
59 views

Klenke's Proof of De Finetti's Theorem

There's a technical problem I ran into when working through Klenke's proof of De Finetti's theorem (Theorem 12.24 on Klenke's Probability Theory: A Comprehensive Course, pages 239-240). The notation ...
user avatar
  • 527
0 votes
1 answer
57 views

a symmetrical die is thrown, then a coin is thrown as many times as the die indicates .

Exercices : a symmetrical die is thrown, then a coin is thrown as many times as the die indicates points. let $X$ be the number of Tails obtained . Determine $E(X)$ My attempts : First it's clearly ...
user avatar
  • 2,135
0 votes
0 answers
15 views

Versions of stochastic processes have the same generated sigma algebra

Let $X_t$ and $Y_t$ be two stochastic processes such that they are versions of each other, viz, $\mathbb{P}(X_t = Y_t) = 1, \forall t$. Is it then so that the sigma albegra generated by $X$ is the ...
user avatar
  • 1
1 vote
0 answers
16 views

Is this Markov process Gaussian ? $Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1]$

Let $(X_t)$ be a real sample continuous stochastic process with density function $f_t$. Let $$Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1].$$ Suppose that $(Y_t)$ is Markov with regard to its own ...
user avatar
  • 2,050
0 votes
0 answers
25 views

To confirm a possible mistake/typo in the book "PROBABILITY AND STOCHASTICS (GTM 261)". Limit arising from a filtration.

In Proposition 4.16, it is written that $\mathcal{F}_{T}=\lim_{n}\mathcal{F}_{T\wedge n}$. Recall that given a sequence of sets $(A_{n})$, if $A_{1}\subseteq A_{2}\subseteq\ldots$, then we denote $\...
user avatar
1 vote
1 answer
22 views

Speed of convergence of empirical probability mass function

I was wondering about the following situation: Suppose we have a random variable $X$ taking values in the finite set $\{1,2,\ldots,k\}$. Let the probability mass function be denoted by $f$. Suppose we ...
user avatar
  • 939
1 vote
2 answers
53 views

If $X$ and $Y$ are independent, is it true that $\mathbb P (X+Y>x | X+Y>0) \geq \mathbb P (Y>x | Y>0)$ for every $x>0$?

$Y$ is a random variable with symmetric distribution around 0 and both random variables are continuous with existing density It seems true, because the random variable $X$ should just bring further ...
user avatar
1 vote
0 answers
23 views

Showing Markov property for "Brownian Motion"

In our lecture, we have defined "Brownian Motion" as a Gaussian process $X$ on $\mathbb{R}_+$ with covariance $C(t,s) = t \land s $, where the " " should indicate that this is ...
user avatar
1 vote
0 answers
29 views

Markov Fields as Markov Chains

I am currently looking through the literature on a topic that has so far appeared only in Georgii's Gibbs measures, chapter 10. The question really is, how do I construct a Markov Chain from a given ...
user avatar
1 vote
0 answers
27 views

Finding stationary distribution of random process

Suppose we are given $x_t, \bar{x_t}, t\in \mathbb{Z_+}$ independent 2-states $\{0, 1\}$ Markov chains with positive transition probabilities. Initial states are $x_0 = 0; \bar{x}_0 = 1$. For which ...
user avatar
1 vote
1 answer
24 views

Central limit theorems and almost sure invariance principles

this is a more general question. Consider a sequence $(X_j)_{j \in \mathbb{Z}}$ of iid real-valued random variables with mean zero and $\mathbb{E}(X_1^2) = 1$ on a probability space $(\Omega, \mathcal{...
user avatar
0 votes
1 answer
46 views

Laplace transform of measure on complex plane

I construct Laplace transform of measure on complex plane,i.e. $F(t)=\int_\mathbb{C}e^{-\lambda t}d\mu(\lambda), t \in \mathbb{R},\mu(\mathbb{C})=1$ and have bounded support. Suppose that $F_n(t)=\...
user avatar
  • 45
1 vote
0 answers
29 views

Can we build an uncountable family of independent random variables with prescribed distributions?

Given a sequence of probability measures $(\mu_n)_{n \in \mathbb{N}}$, I know how to build a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and a sequence of $\mathbb{P}$-independent random ...
user avatar
  • 4,628
0 votes
0 answers
28 views

A notation in measure theory

I was reading about Probability: A Graduate Course by Allan Gut, and I met theorem 3.1 on page 11: Theorem 3.1. Suppose that $A$ and $\{A_n, n ≥ 1\}$ are subsets of Ω, such that $A_n \nearrow A \ (A_n ...
user avatar
1 vote
1 answer
26 views

Conditional density: exact definition

I have trouble understanding the connection of conditional densities for two (continuous) random variables and conditional distributions for measurable sets. Let $\nu$ be a probability measure on $\...
user avatar
  • 215
3 votes
2 answers
537 views

$\sum_{A\in2^\Omega}P(A)=2^{|\Omega|-1}$ for probability space $(\Omega,2^\Omega,P)$ with finite $\Omega$

I'm looking for a combinatorial argument to complete a proof (below) of the following: Claim: If $(\Omega,2^\Omega,P)$ is a probability space with finite $\Omega,$ then $\sum_{A\in2^\Omega}P(A)=2^{|\...
user avatar
  • 12.6k
0 votes
0 answers
46 views

Finding the expected value of two correlated RVs

$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\...
user avatar
  • 1,382
2 votes
2 answers
63 views

$E(h(X_n)) \to E(h(X)).$

Let $X,X_n(n=1,2,\ldots)$ random variable such that $X_n \xrightarrow{a.s.} X$, $E(|X_n|) \leq K < \infty$ $(n=1,2,\ldots)$ and let $h : \mathbb{R} \to \mathbb{R}$ continuos function such that $$\...
user avatar
2 votes
1 answer
22 views

Does a bounded martingale imply a bounded stochastic integral?

Suppose $X$ is a uniformly bounded continuous martingale with $X_0=0$. I am trying to show that $$Y_t:=\int_0^t|X_s|^p\text{sgn}(X_s)dX_s$$ is a true martingale, where $p\geq1$. As $Y$ is a local ...
user avatar
0 votes
0 answers
17 views

How to calculate the Ito formula to a PDE

For a PDE: $$ \frac{\partial u}{\partial t} + Lu+f(x,u,\sigma^{T}\triangledown u) = 0 $$ With $u(x, T) = \Phi(x)$, where $L$ is an operator defined as follows: $$ L\phi = \frac12 \Sigma_{i,j = 0}^{n} ...
user avatar
  • 141
1 vote
0 answers
38 views

Is there a definition of correlation for multiple random variables? [closed]

I know that there is a definition of independence of random variables for finitely many random variables, and this is NOT equivalent to pairwise independence (although it implies pairwise independence)...
user avatar
3 votes
1 answer
60 views

Inequality on expectation of exponential martingale

Let $X$ be a continuous local martingale with $X_0=0$. Define the exponential local martingale $$\mathcal{E}(X)=e^{X-\frac{1}{2}[X]}.$$ For any $p,q>1$, establish the identity $$\mathcal{E}(X)^p=\...
user avatar
1 vote
1 answer
13 views

Is it a problem that we have to *choose* reference measure in Likelihood function?

Suppose we have some parametric model $\{P_\theta \ \colon \theta \in \Theta\}$ and a sample $X$. If we suppose as usual in classical statistics that $P_\theta << \lambda$ for all $\theta \in \...
user avatar
  • 703
0 votes
1 answer
30 views

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
user avatar
  • 1,199
1 vote
1 answer
25 views

$\mathbb{P}\left(X\geqslant\frac{2\alpha}{\lambda}\right)\leqslant \left(\frac{2}{e}\right)^{\alpha}.$

Using $$\mathbb{P}(X\geqslant x)\leqslant e^{-tx}M_{X}(t),\text{ }t\geqslant0,$$ show that in the particular case that $X\overset{\underset{d}{}}{=}\Gamma(\alpha,\lambda)$, $$\mathbb{P}\left(X\...
user avatar
  • 791
2 votes
1 answer
33 views

What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
user avatar
0 votes
1 answer
36 views

Almost sure convergence for lipschitz functions

Let $x_n \to x$($x_n$ sequence of random variables) s.t $\sum \mathbb{E}|f(x_n) - f(x)| < \infty$. For any $f$ Lipschitz and bounded. Then $x_n \to x$ almost sure. My attempt: As series converge, ...
user avatar
2 votes
2 answers
38 views

Hoop Game Throw Ring Toss

I have a problem figuring out what seems a paradoxe. This is a Hoop game in which we throw a skittle (or pole) on a square and flat surface, and then we have to throw (at random) successively $n$ ...
user avatar
  • 303
3 votes
0 answers
47 views

Is recycling samples better than drawing fresh ones?

At a high level, I am wondering if in a sequential process it is better to reutilize samples even if these samples have been used to make past decisions. Let me formalize my doubts in a toy example ...
user avatar
2 votes
2 answers
57 views

Given a random variable $X$, can any random variable $Y$ be "decomposed" as a function of $X$ and $Z$ independent of $X$?

I want to know if given a random variable $X$ on some measure space $(\Omega, \mathscr M, P)$, can any random variable $Y$ on the same measure space be "decomposed" as a function of $X$ and ...
user avatar
  • 5,726
1 vote
0 answers
54 views
+200

Extending a function that gives a value to convex functions to a measure

I am wondering if such a result exists (or similar) and or if there is a "simple" proof. Let $\mathcal X$ be a bounded and closed subset of a topological vector space, let $\Sigma$ be the ...
user avatar
  • 3,965
1 vote
0 answers
26 views

Properties of sum of Poisson processes

Let $N_1$ and $N_2$ be a independent Poisson processes with intensities $\lambda_1=1$ and $\lambda_2=4$. Let $N=N_1+N_2$ and $S_n$ be a moment of $n$ event. I need to calculate the following: $P(N_1(...
user avatar
  • 470
0 votes
0 answers
23 views

Two questions about inhomogeneous Poisson process

Let $N(t)$ be an inhomogeneous Poisson process with a intensity function: $$\lambda(t)=3t+1,\ for\ \ 0\leq t\leq 5,$$ $$\lambda(t)=3\ for\ \ t>5$$ I need to calculate: $E(N(6)-N(3)|N(2)=3)$ $P(N(6)...
user avatar
  • 470
1 vote
1 answer
23 views

Showing that a process is a supermartingale using Ito's formula

Consider a stock with price dynamics $$dS_t=S_t\sigma_tdW_t$$ where $(W_t)_{t\geq0}$ is a Brownian motion and $(\sigma_t)_{t\geq0}$ a bounded and continuous process adapted to the filtration $(\...
user avatar
0 votes
1 answer
23 views

Prove or disprove with $\lim_{n \rightarrow \infty}\ \varphi \ v_{n}$ the existence $V$ with $V_{n}\overset{d}{\rightarrow} V ,n\rightarrow \infty$

Prove or disprove with $\lim_{n \rightarrow \infty}\ \varphi \ v_{n}$ the existence of a random variable $V$ with $V_{n}\overset{d}{\rightarrow} V ,n\rightarrow \infty$ $V_n$ is a sequence of random ...
user avatar
  • 1,005
1 vote
1 answer
53 views

$\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$ from $\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$

Let $X$ be a random variable with characteristic function given by $$\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$$ Determine $\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$. ...
user avatar
  • 791
0 votes
1 answer
23 views

Question on interchanging of random variables with the same distribution inside expectation

Let $T\in\mathbb{N}$ and let $(S_t)_{0\leq t\leq T}$ be such that the increments $S_1-S_0,\dots,S_T-S_{T-1}$ are independent and identically distributed, and let $(\mathcal{F}_t)_{0\leq t\leq T}$ be ...
user avatar
5 votes
2 answers
175 views

Durrett's Probability: Theorem 6.2.6

I am having some difficulty understanding the concept of measure preserving, invariance, and ergodic. Here is a proof from Theorem 6.2.6 in Durrett's Probability: Theory and Examples, 5e (p.338) (...
user avatar
  • 511
1 vote
0 answers
27 views

Are stochastic processes defined on product of probability spaces?

The following question has confused me lately. Suppose that you have a sequence of random variables $\{\xi_n(\omega)\}$ defined on some probability space $\left( \Omega,\mathcal{F},\mathbb{P} \right)$,...
user avatar
1 vote
1 answer
35 views

Determine the weak convergence and applicable the weak limit

$(\mu_n)_{n \in \mathbb{N}} $ is the probability measure, $\lambda$ a Lebesgue-measure on $(\mathbb{R}, \cal{B}\mathbb{(R)})$ $$(\mu_n)_{n \in \mathbb{N}} = f_n \lambda, f_n(x) = \sqrt{\frac{n}{2 \pi}...
user avatar
  • 1,005
2 votes
1 answer
46 views

A statement about finite Markov chains

The following quantity $\tilde\pi$ is defined in the textbook Markov chains and mixing times by David A. Levin. Here $\tau_z^+ = \min \left\{t\geq 1| X_t = z\right\}$. Let $z \in \mathcal{X}$ be an ...
user avatar
1 vote
0 answers
19 views

Computing transition probabilities in continuous time Markov chain

How do we compute the transition probabilities in a continuous time Markov chain? Supposing $h$ is sufficiently small then how would I compute $p_{i,j}(h)$, I am aware of the relation to the generator ...
user avatar

1
2 3 4 5
787