All Questions
9 questions
0
votes
0
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41
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Formalization of sample convergence
Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that
$$X_i = \sigma_i z_i,$$
where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
- 33
27
votes
5
answers
3k
views
How to show a function converges to 1
Consider the following recurrence relation in two variables:
$$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$
for positive integers $a$ and $b$,
with the boundary conditions $f(0,b)=0$ ...
- 31
6
votes
2
answers
774
views
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
- 1,986
1
vote
0
answers
96
views
Limit of alternating sum of factorial moments which diverge
Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...
- 640
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 128k
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 128k
1
vote
1
answer
108
views
Maximum of the periodogram of a truncated sequence
Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...
- 3,968
3
votes
1
answer
105
views
How to show monotonocity and the limit? [closed]
Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...
- 128k