Linked Questions

2
votes
1answer
215 views

Does $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ imply independence of $X$ and $Y$? [duplicate]

It shouldn't, but I am blanking on a counterexample. ETA: Note that the $t$ is shared on both sides - which differentiates this from this question. Similarly $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ implies ...
0
votes
0answers
57 views

Do characteristic functions characterize the independence of random variables? [Solved] [duplicate]

It is well known that the probability density function characterizes the independence of random variables in the following sense. $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$ where $f$ is ...
12
votes
2answers
5k views

A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
6
votes
2answers
1k views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
2
votes
2answers
959 views

Show that if two random variables sequences are pairwise independent then the limits are independent, too.

Two sequences $X_1, X_2, \ldots, Y_1, Y_2,\ldots : (\Omega, \mathcal{F},\mathbb{P}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ of real random variables such that $\forall n \ X_n, Y_n $ are ...
5
votes
1answer
898 views

X,Y,Z are mutually independent random variables. Is X and Y+Z independent?

X,Y,Z are mutually independent random variables. Is X and Y+Z independent? Please, give me a hint how to prove it?
4
votes
1answer
441 views

Does $E[e^{it(aX + bY)}]=E[e^{itaX}]E[e^{itbY}]$ for every $a,b\in\mathbb{R}$ imply that $X$ and $Y$ are independent?

Let $X, Y$ be two random variables such that for every $\alpha, \beta \in \mathbb{R}$, $$E[e^{it(\alpha X + \beta Y)}]=E[e^{it\alpha X}]E[e^{it\beta Y}]$$ for all $t\in\mathbb{R}$. Does it follow ...
2
votes
1answer
701 views

Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
2
votes
3answers
423 views

Why are these two Poisson-processes independent?

I have two poisson-processes, I have seen a mathematical proof that they are independent, and offcourse they must be independent since the proof is in several textbooks. But logically I can not ...
3
votes
2answers
104 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
2
votes
1answer
193 views

Verifying a Brownian motion through the Laplace transform

Let $X(t)$ be a continuous stochastic process and $\mathcal G(t)$ be the $\sigma$-algebra generated by $\{X(\tau) : \tau\leq t \}$. Suppose that for any $0\leq s\leq t$ and $\lambda\in\mathbb C$ ...
4
votes
1answer
62 views

Why does $\mathbb{E}(F(X) 1_A) = \mathbb{P}(A) \mathbb{E}(F(X))$ imply the independence of $X$ and $A$?

I was trying to understand the Markov property of Brownian Motion and in one of the proofs the author claims that the following result Let $\mathcal{F} \subseteq \mathcal{A}$ be a $\sigma$-algebra ...
1
vote
1answer
156 views

$P(\tau_k<\infty)=q^k$ for hitting time $\tau_k =\inf\{t; X_t=k\}$ of asymmetric random walk

Consider the random walk where $X_t=\sum_{i=1}^t Y_i$, $Y_i$'s are iid and take $\pm 1$ with probabilities $p$ and $1-p$ respectively, where $0<p<0.5$. Define stopping time $\tau_k=\min\left\{t:...
1
vote
0answers
98 views

Some basic questions related to independence of random variables

I attend a lecture about Stochastic Processes even though I have not studied mathematics and some of the basics in probability theory are missing. So I hope you can help me with the following ...
1
vote
1answer
57 views

independence between rv and $\sigma$-algebra

assume $\mathcal{A}$ is a $\sigma$-algebra and $\xi$ is a r.v.  and $\forall x\in \mathbb R$  $$\mathbb{E}\left[e^{ix\xi} \mid \mathcal{A}\right] = \mathbb{E} \left[ e^{ix\xi} \right]\tag{*}$$ try to ...

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