Tagged Questions

2
votes
1answer
114 views

Conditional expectation and martingales

I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$. We ...
1
vote
2answers
40 views

Must the sequence $X_n$ converge to $0$ in probability?

Let $X_1, X_2,\dots$ be a sequence of random variables with $\lim_{n\to +\infty} E[|X_n|] = 0$. Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
2
votes
1answer
112 views

Independence of two limits

Let $(X_n)$ and $(Y_n)$ be two sequence of random variables. $(X_n)$ and $(Y_n)$ are independent to each other. If $(X_n)$ and $(Y_n)$ have limits in distribution. $(X_n)$ tends to $X$, and $(Y_n)$ ...
2
votes
0answers
68 views

Alternative way of showing convergence of central moments

I woud like to show for a random variable $X$ $${1\over n} \sum_{i=1}^n (X_i-\bar{X}_n)^q\to E(X-EX)^q \quad (\text{convergence in probability)}$$ My approach was to define $Y_i:= (X_i-\bar{X}_n)$ ...
0
votes
0answers
31 views

a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?

Any example that a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
0
votes
1answer
141 views

Almost sure convergence of random variables

Assume that $X_n$ are independent (but not necessarily of the same distribution) and that $Var[X_n]>0$ for all $n$. We know that $$\frac{X_n-E[X_n]}{n}\to 0 \textrm{ almost surely as ...