# Tag Info

• 992
Accepted

### Prove the existence of $g$ such that $\mathbb E (e^{aX_t} ) = e^{g(a) t}$ for a Compound Poisson Process $X_t$

By the Tower Property: $$E[e^{aX_t}] = E[E[e^{a(Y_1 + \cdots + Y_n)}] | N_t = n]$$ And since $(Y_i)$ are iid, we simplify the above to: $$E[E[(e^{aY_1})]^n | N_t = n]$$ As $Y_i$ is independent of $N$, ...
• 4,236
1 vote

### Independence assumption for interarrival time

You have to keep in mind the separation between a physical process and a model. We observe some process in the real world (customers arriving at a server) and want to construct a model that helps us ...
• 1,036
1 vote

### Independence of Sample Mean and Sample Variance

Yes it is a characterization for the normal distribution shown first by Geary (1936) that the sample mean and sample variance are independent if and only if the population distribution is normal. ...
• 6,785
1 vote

### Does Independence hold for $\sigma$-Algebras Generated by Disjoint Subsets of an independent Sequence

I think I understand how to prove it now. Please let me know, if my reasoning is correct. Big thanks to @IzzakvanDongen. Let $\mathcal{A}$ and $\mathcal{B}$ denote the sets of all possible finite ...
• 243

I will show the argument when the subsets of $\mathbb{N}$ are finite. Let $A=\{a_1,...,a_n\}\subset \mathbb{N}$. Let $(X_n)_{n \in \mathbb{N}}$ be random variables. Then, $(X_{a_1},...,X_{a_n}):\Omega\... • 15.5k 4 votes Accepted ### Prove that$Z_n$is independent of$(Y_{i,n})_i$First, note that$Z_n$is$\sigma(Z_{n-1}, Y_{1,n-1}, Y_{2,n-1}, \ldots)$-measurable. By induction on$n$, this shows that$Z_n$is$\sigma(Y_{i,j} : i\geq 1, 1\leq j < n)$-measurable. It now ... • 21.7k 0 votes ### Independence vs. Measurability Assuming that X is non-trivial, we can find that$E[X|\mathcal{G}]=EX$, if X is$\mathcal{G}$-measurable, and$E[X|\mathcal{G}]=X$, if X is independent to$\mathcal{G}$. Thus, we can conclude that if ... 2 votes Accepted ###$XY$are independent so are$f(X)$and$f(Y)$if$f$is continuous It's easier than that. If$X_1$and$X_2$are independent, if we set$Y_i = f(X_i)$then for any two measurable sets$A,B$we have$$P(Y_1 \in A, Y_2 \in B) = P(X_1 \in f^{-1} (A), X_2 \in f^{-1} (B)) ... • 13.2k 0 votes Accepted ### Equivalent definition of independent increments of a stochastic process. The problem becomes easy if you assume$X_0=0$surely, as you indicated in your comments. The main idea is to consider the vectors$(X_0, X_{t_1}, ..., X_{t_n})$and$(X_0, X_{t_1}-X_0, ..., X_{t_n}-...
• 24.3k
As a toy example, consider a Yule process $(Y_t, t\geq 0)$: It's law can be described as follows: $Y_0=1$, a.s. If $Y_t=n$, then the transition to $n+1$ happens at rate $n$. This is very similar to ...