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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
21 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in ...
1
vote
0answers
58 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
0
votes
1answer
14 views

Using probability or moment generating functions to find the distribution when given the distribution with random parameters

Problem: Given $X|M=m \sim $ Poi(m) with $M\sim $ Exp(1). Find the unconditional distribution of X where M is the random parameter. I want to solve this using P.G.F $g(t)$ and/or M.G.F $\psi(t)$. ...
0
votes
1answer
32 views

Use the delta method to find the distribution of $Z_n$

Let $\overline{X}_n=\overline{X}$ the sample mean such that $\sqrt{n}\overline{X}_n\rightarrow^D N(0,1)$ where $\rightarrow^D$ means converge in distribution. Use the delta method to find the ...
0
votes
1answer
25 views

Simple inequality of tails of random variables

Let $a>0$, $X$ some random variable and $\mathbb{E}[X]=\mu<t$. I was trying to prove the following simple inequality: $$ \textrm{Pr}\left[|X-t|\geq a\right]\leq\textrm{Pr}\left[|X-\mu|\geq ...
0
votes
2answers
27 views

Relative entropy (KL divergence) of sum of random variables

Suppose we have two independent random variables, $X$ and $Y$, with different probability distributions. What is the relative entropy between pdf of $X$ and $X+Y$, i.e. $$D(P_X||P_{X+Y})$$ assume all ...
3
votes
1answer
43 views

Is it almost impossible for the values of continuous random variables to lie on a plane?

Let $X: \Omega \to \mathbb{R}^d$ be a random variable with density $f$ (the pushforward measure on $\mathbb{R}^d$ is absolutely continuous). Let $x_1, \dots, x_{d}$ be the set of values of $d$ i.i.d. ...
0
votes
0answers
20 views

Convergence of $\prod\limits_{t\in\mathbb{N}}P(|X_t|<\lambda^t)$, where $\lambda>1$

Let $\{X_t\}_{t\in\mathbb{N}}$ be an independent sequence of continuous random variables on the real line. Let $\lambda>1$. I am interested in the quantity $$ P(\forall t\in\mathbb{N},\,|X_t| \le ...
1
vote
1answer
26 views

Drawing Random Variables

What does it actually mean when we draw a number from a given distribution? To elaborate on what exactly I'm trying to ask: if I generate a vector of numbers from the standard normal distribution on ...
0
votes
1answer
28 views

Find the PDFs of X+Y [closed]

If $X$, $Y$ have joint PDF, $$f(x,y) = \begin{cases} 1 & \text{if }~0 \le y \le 1,~2y\le x \le 2 \\[1ex] 0 &\text{if elsewhere} \end{cases}$$ Find the PDF of $X + Y$. Hi, I am stuck on this ...
2
votes
1answer
49 views

application of Birkhoff Ergodic Theorem

Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded ...
2
votes
1answer
91 views

Independence of RVs conditioned on an event

Assume that we know that $X_1 \in R^n$ and $X_2 \in R^n$ are independent conditioned on $Y=y$, that is, for any $E_1 \subset R^n$ and $E_2 \subset R^n$, $P(X_1 \in E_1, X_2 \in E_2 \vert Y=y) = ...
-1
votes
1answer
48 views

A question of determining when the entropy is maximum.

Y ={ 1, 2,...,r} We are given that X is the set of two sided sequences with entries from Y and T is the two sided shift on X, and m is a T invariant probability measure on X. If $p_i = m(\{x ...
2
votes
2answers
62 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
0
votes
0answers
40 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
0
votes
0answers
9 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $y_i=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those ...
0
votes
1answer
27 views

Do these integrals converge to 0?

Assume that you have a probability space $(\Omega,\mathcal{F},P)$. And you have a positive random variable $Z$, with $E[Z] =1$. You can then define a new probability space $(\Omega,\mathcal{F},P')$, ...
0
votes
0answers
47 views

Bivariate normal distribution in polar coordinates with unknown correlation between the variables.

Consider the problem of finding the dist. of $\theta$ after changing a two dimensional normal distribution to polar coordinates where both variables are standard. Using transformation theorem I get; ...
2
votes
1answer
16 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, ...
0
votes
1answer
18 views

What is the probability distribution of the following random variable?

Let $A^n$ and $B^n$ be independent random variables taking values in $\{0, 1\}^n$. Let $Y^n = A^n + B^n$ (Hence, taking values in $\{0, 1, 2\}^n$). How can we express the distribution of $Y^n$ in the ...
0
votes
1answer
21 views

Calculating convolution of binomial distribution using moment generating function

I have two independent random variables $X_{1}, X_{2}$ on the same probability space. $X_{1}$ is bin bin(n, p) and $X_{2}$ ís bin (m, p) with n, m natural numbers and p in the interval [0,1]. I need ...
0
votes
0answers
16 views

Relation between Kolmogorov Zero-One Law and Random Graphs Zero-One Laws?

I know of Zero One laws for Random graphs (such as those concerning monotonic or first-order-logic properties). I also know about Kolmogorov's zero one law for tail sigma algebras. Apart from the ...
1
vote
0answers
17 views

What is the chance that two random binary random variables are independent?

Consider the space of probability distributions on 4 letters. Now all the probability distributions on four letters do not represent a distribution of two independent binary random variables. But if ...
0
votes
1answer
45 views

non negative super martingale

Let $(X_n)_{n\geq0}$ be a non-negative supermatingale and $T = \inf\{n \geq 0 : X_n = 0\}$. Show that on the event $\{T < \infty\}$, $X_{T+n} = 0$ for all $n \geq 0$ a.s. My approach: $0 \leq ...
1
vote
1answer
45 views

Show $\mathbb{E}(\sum_{i=1}^n(X_i-(\frac{1}{n}\sum_{i=1}^{n}X_i))^2)=(n-1)\mathbb{V}(X)$

Let $X_1, X_2, \ldots$ be a sequence of i.i.d random variables with finite variance and $M_n=\frac{1}{n}\sum_{i=1}^{n}X_i$. We need to show that ...
1
vote
1answer
22 views

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $f_{X,Y}(x,y) = 6x$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $$f_{X,Y}(x,y) = 6x$$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$ I have the solution to this, but I don't understand it completely. Can some one ...
0
votes
0answers
17 views

What are the differences in linearity in Non-stochastic and Stochastic Regression?

I have been confused with the differences between stochastic and non-stochastic explanatory variables for a while. I was able to write down some of my understanding and seek approval or comments about ...
0
votes
1answer
23 views

Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
1
vote
0answers
25 views

Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$

I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it. Proposition: $\lim \inf ...
1
vote
1answer
42 views

Proof that $E(F_X(X/\sigma))=\frac12$ for every positive $\sigma$

Define $X$ to be continuous random variable symmetric about zero with cdf $F_X$ and let $\sigma > 0$ denote a constant. Now show the following: $$ E\left[F_X\left(\frac{X}{\sigma}\right)\right] = ...
1
vote
0answers
37 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ...
1
vote
0answers
31 views

Proof that $\mathcal L^2 \supsetneq \mathfrak B^2$.

Let $\mathfrak L^d$ be the $\sigma$ -algebra of all Lebesgue-measurable subsets and $\mathfrak B^d$ the one of the Borel sets in $\mathbb R^d$. I want to prove that $\mathfrak L^2 \supsetneq \mathfrak ...
1
vote
1answer
19 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. ...
1
vote
0answers
41 views

Mixing convergence

Given a process $X_n \xrightarrow{d} X$ on some probability space $(\Omega,\mathcal{A},P)$. If for every $B \in \mathcal{A}$ it holds, that $$ \lim_{n\rightarrow \infty} P(X_n\in A,B)=P(X\in A)P(B) $$ ...
1
vote
1answer
23 views

On characteristic function of multivariate normal distribution

I cant make sense of (4.2), putting $Z=1$ should only establish one point not the whole function right, hence I dont see how Thm 4.1 is "proved"? Second RHS is not even a funtion of $t$. Anyone have ...
2
votes
0answers
37 views

Dirichlet Problem in Stochastic PDE Section of Probability Textbook.

I recently started learning about stochastic calculus and stochastic PDEs, and I don't know where to begin with some of the problems in my textbook. Any help with the following or a push in the right ...
1
vote
2answers
34 views

Problem in “Probability - An Introduction” by Grimmet and Wesh

A fair die having two faces coloured blue, two red and two green, is thrown repeatedly. Find the probability that not all colours occur in the first $k$ throws. My attempt: Denote by $G_k$ the number ...
1
vote
0answers
16 views

Show the sample mean converges to minus infinity when ${X_n}$ are i.i.d. and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$

Suppose ${X_n}$ are iid and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$. Show that if $S_n := \sum_{j=1}^nX_n$ then $\frac{S_n}{n} \rightarrow -\infty$ almost surely. A hint in my book says to use ...
1
vote
1answer
26 views

CDF for a random variable that is neither discrete nor continuous

We flip a fair coin. Then, if the result is tails, we stop. If it is heads, we flip a second time and then stop. Let $X$ be the number of heads from the flip(s). If $X = 0$, let $Y = 0$. If $X = 1$ ...
0
votes
1answer
40 views

Assume that $X,X_1,X_2,…$ are iid with characteristic function $\phi(t)=\mathbb E[e^{itx}]$, and let $S_n = X_1 + X_2 + X_3 + …$.

Assume that $X,X_1,X_2,...$ are iid with characteristic function $\phi(t)=\mathbb E[e^{itx}]$, and let $S_n = X_1 + X_2 + X_3 + ...$. (a) For a random variable $X$, $X$ and $-X$ have the same ...
0
votes
1answer
63 views

Forward price in Black Scholes Model

Recall that a forward contract on $S_T$ contracted at time $t$, with time of delivery $T$, and with forward price $f(t; T, S_T)$ can be seen as a contingent T-claim $X$ with payoff: $$ X = S_T - f(t; ...
3
votes
1answer
70 views

Using the martingale central limit theorem

Suppose a box has $2n$ tickets half of which are labelled $+1$ and half $-1$. Labeling the draws without replacement by $X_1, ...$, define $S_m = X_1 + ... + X_m$. For any $t \in (0,1)$ ...
0
votes
0answers
14 views

Fourth moment of a stationary GARCH process

I have a martingale difference series $(Z_t)$ with respect to a given filtration $(\mathcal{F}_t)$, i.e. $E[Z_t \mid \mathcal{F}_{t-1}] = 0$. Furthermore, $E[Z_t^2 \mid \mathcal{F}_{t-1}] = 1$. I also ...
1
vote
1answer
24 views

Equal distribution only for finite dimensional distributions

Two processes $(X_t)_{t \in T}$, $(Y_t)_{t \in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions, i.e., for all $t_1$, $t_2$, $\ldots$, $t_n$, ...
0
votes
1answer
72 views

optimal stretegy

Imagine the following game. A player rolls a die. If the player rolls a $6$ the player wins no money. Otherwise, the player may either quit the game and win $k$ dollars, where $k$ is the roll of the ...
2
votes
0answers
26 views

Is every discrete martingale a time-changed simple random walk?

While going through the book by Revuz and Yor titled 'Continuous Martingales and Brownian Motion', I came accross the notion of time change. In a nutshell, if X is a stochastic process and C is an ...
1
vote
2answers
28 views

Which definition is correct for a geometric random variable?

Is it The number of failures BEFORE the first success OR The number of trials required to get a first success? Also, if I was to work out the expected value of a geometric random variable, say $p ...
0
votes
0answers
13 views

Doubts in Ignatov's Theorem proof (Sheldon M. Ross Book)

Good morning, I am working in Ignatov's theorem and I have a doubt regarding the proof which can be seen in Sheldon M. Ross book (Introduction to probability models). We have a sequence of ...
0
votes
0answers
21 views

Almost sure convergence on conditional probability

Say we have a compact space $X$ and a probability measure $P$ on $X$. Assume that we know that for some event $A$ the sequence $f_n(x)\rightarrow f_\infty(x)$ converges a.s. in $Q$ which is the ...
0
votes
0answers
20 views

How to Prove that a (Centered) Gaussian Process is Markov if and only if this Equation Holds?

A centered Gaussian process is Markov if and only if its covariance function $\Gamma: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ satisfies the equality: ...