All Questions
34
questions
3
votes
1
answer
889
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Weak convergence: equivalence of definitions
Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent?
Def 1 $(X_n)_{n\geq 0} \...
- 200
2
votes
1
answer
330
views
Weak convergence using characteristic functions
This problem is from Billingsley's "Probability and measure" book.
Let $a_n \to a$, $b_n \to b$ and $\{X_n\}$,$X$ be a sequence of random variables such that $X_n \to^w X$ (weak convergence). Prove ...
- 10.3k
15
votes
1
answer
5k
views
Uniform convergence and weak convergence
Assume $F_{n},F$ are distribution functions of r.v.$X_{n}$ and $X$, $F_{n}$ weakly converge to $F$. If $F$ is pointwise continuous in the interval $[a,b]\subset\mathbb{R}$, show that
$$\sup_{x\in[a,b]}...
- 279
1
vote
1
answer
408
views
Product of a sequence tending to 0 and a sequence of random variables converging in distribution
Let $X_n$ be a sequence of random variables and suppose $X_n \rightarrow X$ in distribution. Let $a_n$ be a sequence of constants with $a_n \rightarrow 0$. Must $a_n X_n \rightarrow 0$ almost surely? ...
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