Questions tagged [probability-theory]
For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.
41,756
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If $B$ is a Brownian motion independent of $X$, then $B$ is still a Brownian motion under the regular conditional probability given $X$
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $B$ be a one-dimensional Brownian motion. Let $X, Y:\Omega\to \mathbb R$ be random variables.
Let $\nu: \mathbb R \times \mathcal F \...
- 13.6k
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1
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27
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Why do singleton events imply sets in multiplicative but not in additive probability?
Let $a \geq 0$ and $0\leq b \leq 1$ and $M, N$ be two appropriate conditioning events such that, for all singletons $y = \lbrace y \rbrace$ in the sample space $Z$ and all subsets $Y$ of $Z$, the ...
- 41
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Different measurability of Hilbert-space valued random variable
My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...
- 12.7k
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1
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18
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Using derivative as probability density function
Let be $X$ an absolutely continuous random variable. Then, we know that there exists a probability density function (pdf) $f_X$ such that its cumulative distribution function (cdf) $F_X(x)$ can be ...
- 3,685
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1
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30
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understand proof of strong law of large numbers (using reverse martingale)
I am reading the proof of strong law of large numbers using reverse martingale here. There are several points about conditional expectation that I do not understand. They all seem pretty intuitive but ...
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1
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1
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31
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How can I show that $(\|B_t\|^2-dt)_{t\geq 0}$ is a martingale?
Let $B$ be a $d$-dimensional $(\Bbb{F}, \Bbb{P})$-Brownian motion where $\Bbb{F}=(\mathcal{F}_t)_{t\geq 0}$. Then consider $X:=(X_t)_{t\geq 0}=(\|B_t\|^2-dt)_{t\geq 0}$. I want to check that it is a ...
- 51
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1
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60
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If $\mathbb EX=0$ and $\mathbb E|X| < \infty $ implies finite Variance?
I think, I am totally wrong, but
if $\mathbb E|X|$ is finite then,
$$\text{Var}|X|=\mathbb E|X^{2}|- (\mathbb E(X))^{2}=\mathbb E|X^{2}|< \infty$$
Considering $\mathbb E|X|< \infty$ implies $\...
- 163
3
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2
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128
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Probability and Random Variables.
Hi, I was trying to understand this example in the book. In the first part of the question, We've to find p.d.f (probability density function). For that, we take the derivative of the given ...
0
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0
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21
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`Cropped' $\max_{0\leq k\leq n}S_k - S_n$, where $(S_n)_{n\in \mathbb{N}}$ is a simple random walk, has the same distribution as $|S_n|$.
Let $S_n$ be a simple random walk with $S_0=0$ and $X_n:=\max_{0\leq k\leq n} S_k - S_n$ is a non-decreasing process which is zero when $n=T_n:=\max\{k\leq n: \sup_{0\leq k\leq n} S_k = S_n\}$.
I want ...
- 433
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1
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42
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Can we characterize conditional independence by conditional probability?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $(S, d)$ be a Polish space and $\mathcal S$ its Borel $\sigma$-algebra.
Let $\mathcal A$ be a sub-$\sigma$-algebra of $\mathcal F$. ...
- 13.6k
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1
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23
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Young's inequality in probability theory
The following is the standard version, in two equivalent statements:
If $a \geq 0$ and $b \geq 0$ are nonnegative real numbers and if $p > 1$ and $q > 1$ are real numbers such that $\frac{1}{p} ...
- 41
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21
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Conditional independence and conditional probability
I'm trying to make clear about the notion of conditional independence. Could you have a check on my below understanding?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $\mathcal A, ...
- 13.6k
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0
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19
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Interchanging expectation and integration with a collection of random integrands
Suppose one has a collection of i.i.d. random functions $\{f(\cdot,t):t\in\mathbb R\}$, where we write $f$ for the common distribution. In a probability application, what I need is to interchange ...
- 3,013
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0
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11
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Integral with respect to the Brownian sheet/Wiener Field [duplicate]
Let a Brownian sheet/wiener field, $W\left(t_1, \ldots, t_d\right)$, for some $d \in \mathbb{N}$ ( see this : https://encyclopediaofmath.org/wiki/Wiener_field for a definition).
For the $d=1$ case, we ...
- 65
1
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1
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49
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Probabililty of sums of gaussians
Let $X\sim N(\mu_1, \sigma_1)$ and $Y\sim N(\mu_2, \sigma_2)$. Consider an arbitrary $r\in\mathbb{R}$. How can I compute the probability of $\mathbb{P}(X \leq r \leq X + Y)$?
My idea: One could write ...
- 106
6
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0
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91
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When does the power series $\sum_{n=1}^\infty a_nx^n$ have infinitely many zeros in $(0,1)$?
Let $f(x)=\sum_{n=1}^\infty a_nx^n$ be a power series, where $a_n=\pm 1$. I have encountered a probability question, saying that $(a_n)_n$ is a sequence of i.i.d. Bernoulli random variables, with $\...
0
votes
1
answer
66
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On the Cauchy-Schwarz inequality: can there be something in between?
For any couple of random variables $X$ and $Y$,
$$
\displaystyle |\mathbb{E} (XY)|^{2}\leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2})
$$
Is it possible to put an intermediate bound, like
$$
\displaystyle |\...
- 41
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0
answers
21
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On the proof of Ito Isometry using partial sums
This post on MSE provides a nice derivation, but I don't understand this crucial step
$$\begin{align}&\mathbb E\left[\sum_{i≠ j}K_iK_j(M_{t_{i+1}} - M_{t_i})(M_{t_{j+1}} - M_{t_j})\right] \\
=&...
- 41
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1
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Mathematics Behind Coincidence of Probabilities (Magic the Gathering Example)
In Magic the Gathering, a player must decide how many land cards to include in his or her deck. Drawing too few at the beginning of a game is called mana screw. Drawing too many at the beginning of ...
- 1,554
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1
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21
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Mean square convergence of the squares
Let $X_n$ be a sequence of random variables converging in mean square to $X$, that is,
\begin{equation}\tag{1}
\lim_{n\to\infty}\mathbb{E}\Big(\big|X_n-X\big|^2\Big)=0
\end{equation}
or, in short-hand ...
- 41
1
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1
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28
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An inequality about the 2-Wasserstein distance
Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
- 6,187
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1
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48
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If a sequence of random variables converges in probability to $0$, will the sequence still converge to $0$ when multiplied by $\sqrt n$
Assume we have a sequence of random variables $X_1,X_2...$ such that the sequence converges to $0$ in probability. Under what general conditions will the sequence still converge in probability to $0$ ...
0
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0
answers
28
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Ito diffusion crosses initial value infinitely many times?
Consider the stochastic process $Y_t=y+\int_0^t\sigma(Y_s)dW_s$ where $W$ is a standard Brownian motion. It is well known that if $\sigma(y)=\sigma >0$ constant we have that $Y_t$ crosses $y$ ...
- 12.3k
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16
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Gaussian concentration to upper bound the variance of the norm of a gaussian vector
I want to prove the following:
Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$.
Then:
$ Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2 $
This is ...
- 3
1
vote
1
answer
42
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Calculating Prize Line Expectation Part 2
Thanks in advance for any help.
Yesterday a very helpful member called @joriki answered my original question on this and that conversation came to a conclusion as a result. I have a second part that ...
- 133
1
vote
1
answer
35
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Time interval between the first and the last maxima of random walk $\{S_k\}_{k=1}^n$ as $n\to\infty$
Let $S_n:=\sum_{i=1}^n X_i$ where $\{X_i\}$ are IID with $\mathbb{E}X_i = 0$ and $\mathbb{E}X_i^2=1$ for all $i\in \mathbb{N}$.
We define stopping times: $\tau_n:=\inf\{1\leq k\leq n: S_n = \max_{1\...
- 433
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0
answers
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Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity
I am wondering about how to deal with a limit of the following form
$$ \lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$
with the function $a_n(t)$ and the sum being dependent on $n$.
Consider the ...
- 33
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0
answers
52
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Proof that given a continuous random variable $X$ with density function $f(x)$ then $\mathbb E[g(X)] = \int_{-\infty}^{\infty} f(x)g(x)dx$.
I have tried a million things and I just can't seem to crack this.
To repeat it the question is, given a continuous random variable $X$, with density function $f(x)$ prove that:
$$
\mathbb E[g(X)]=\...
- 102
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votes
1
answer
36
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Expectation for an $x_i$ in a permutation.
Any tips on how to approach the next steps of this problem?
$X$ is a set of $N ≥ 2$ distinct numbers and let $x_1x_2\cdots x_n$ be a permutation of $X$. For
$i = 2, 3, . . . , n$ we say that position $...
- 21
1
vote
1
answer
38
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MLE for the sum of a normally distributed variable and constant after a specific time [closed]
We start off with a normally distributed random variable $X$ with known $\mu=100$ and $\sigma^2=1$, and after $\vartheta$ days, a constant $1$ gets added to the value each day. Given $X_1,...,X_n$ ...
0
votes
1
answer
33
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In what way is a Gaussian process a distribution over function space?
Gaussian processes are generally introduced as families of rvs where all finite vectors are multivariate normal. However, they are also described sometimes as "distributions over functions." ...
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1
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The two random variables $X$ and $Y$ have the following common distribution
EDIT: here is the formatting of the problem. I already solved a) and b), I attempted to solve c) and struggle with d)
$$
\begin{array}{c|lcr}
X \setminus Y & 0 & 1 \\
\hline
-1 & 0 & ...
- 337
1
vote
1
answer
21
views
Let $M$ be a continuous martingale, $r >0$, and $\tau := \inf \{t \ge 0 : \langle M \rangle_t \ge r \}$. Then $M_\tau$ is square-integrable
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous martingale w.r.t. $\mathcal G$ such that
$$
(\...
- 3,892
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1
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Show that $T$ is a stopping time
Show that $$T = \inf \left \{n \geq 0|X_n \in \left \{0, N \right \} \right \}$$ is a stopping time with respect to $\mathcal{F}_n=\sigma(X_0,...,X_n)$ for $n\geq 0$.
I am fairly new to stopping times....
- 23
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votes
2
answers
36
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Given this weak property is it possible to demonstrate that the difference of expected value is negative?
Let's assume that we have $X,Y$ as random variables and we have as hypothesis that
$$X-\mathbb{E}_x \leq Y-\mathbb{E}_y$$
where $\mathbb{E}_x$ is the expected value of x.
Is it possible to demonstrate ...
- 31
0
votes
1
answer
28
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Convergence of ratio of Big O-terms in probability
Say we have constants $v \in \mathbb{R}$ and $w >0$. Assume that $a_n \to 0$ and $b_n \to 0$ as $n \to \infty$. I wish to show that
$$
\frac{v+O_p(a_n)}{w+O_p(b_n)}=\frac{v}{w}+O_p(a_n)
$$
where we ...
- 273
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votes
2
answers
33
views
Let $X_n \to X$ and $Y_n \to Y$ in probability such that $B \subset \{X_n = Y_n\}$ for all $n$. Is $B \setminus \{X=Y\}$ a null set?
Let $(\Omega, \mathcal F, \mathbb P)$ be a complete probability space. Let $X, Y, X_n, Y_n:\Omega \to \mathbb R$ be random variables such that $X_n \to X$ in probability and $Y_n \to Y$ in probability....
- 3,892
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1
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Calculating Expected Increase In Prize Line Count
Hello and thanks in advance.
I'm trying to solve a problem I'm having with a pattern matching game.
The game's board consists of a 5x5 grid of numbers where column 1 (on the far left) contains 5 ...
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votes
1
answer
25
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Intuition of conditioning on future observation in brownian motion?
I've been learning about Brownian motion and have verified a result that for a Brownian motion process $\{B_t : t \geq 0\}$, $B_0 =0 $ and $0 \leq s < t$,
$$
B_s | B_t \sim \mathcal{N}\left(\frac{s}...
- 3
-1
votes
0
answers
25
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Finding the CDF and the PDF of random variable $ Y=C\sqrt{r^2−B^2}^D$
For a random variable given as $R$ with pdf $$f_R(r) = \frac{12}{73} \left(\frac{27r}{M^2}-\frac{35r^3}{M^4}+\frac{8r^5}{M^6}\right)$$
I need to find pdf and cdf $$ Y=C(\sqrt{r^2−B^2})^D$$
where $C , ...
- 1
0
votes
0
answers
52
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How to fill in this proof of Jensen's inequality for conditional expectation?
Recall Jensen's inequality for conditional expectation: $X$ an integrable (or nonnegative), real-valued, $\mathfrak{F}$-measurable random variable. $\varphi: \mathbb{R} \to \mathbb{R}$ convex. $ \...
- 151
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votes
0
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22
views
$\mathfrak{F}$ contains an even number of subsets of $\Omega$
Show that, if $\Omega$ is a finite set, then $\mathfrak{F}$ contains an even number of subsets of $\Omega$.
Proof: Let $f:\mathfrak{F}\rightarrow\mathfrak{F}$ such that $f(A)=\Omega\setminus A,\text{ }...
- 965
1
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1
answer
49
views
How to find a probability given other probability functions?
I have the following problem: consider 4 discrete RVs $a,b,c,d$ with $a, b$ independent. The following pmfs are known: $p(a)$, $p(b)$, $p(c|a,b,d)$, $p(d|a,b,c)$ for every value of $a,b,c,d$. Find $p(...
- 1,092
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votes
1
answer
37
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Markov chains - train ride
Problem:
A country has $m+1$ cities $(m\in\mathbb{N})$, one os which is the capital. There is a direct railway connection between each city and the capital, but no tracks between any two 'non-capital' ...
- 385
1
vote
1
answer
61
views
How can I prove this equivalent relation about measurable random variables?
How can I prove “ random variable $\alpha$ is $\mathcal{F}\vee \sigma(E_k,k=0,1,…,N)$-measurable” if and only if “$\displaystyle{\alpha=\sum_{k=0}^N} \alpha_k\mathbf{1}_{E_k},\ $ where for any k, $\...
- 21
3
votes
1
answer
74
views
Distribution of pre-transformed Gaussian Random variables
Let $f,g:\mathbb{R} \to \mathbb{R}$ be monotonic and differentiable functions, and suppose that $(X, Y)$ is a random vector such that
$$
(f(X),g(Y))^T \sim N (\mu, \Sigma),
$$
where $\mu = (\mu_1, \...
- 5,694
0
votes
1
answer
31
views
Partial order relation on $\mathcal{M}_{1,c}(\mathbb{R})$.
Let $\mathcal{M}_{1,c}(\mathbb{R})$ the space of Borel probability measures on $\mathbb{R}$ with compact support. There is a partial order relation on $\mathcal{M}_{1,c}(\mathbb{R})$:
$$\mu\succeq \nu ...
- 57
0
votes
0
answers
31
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On the autocovariance function of progressively measurable processes
Suppose I have a progressively measurable process $G_\omega(t)$ (adapted process + càdlàg paths), which satisfies
$$\tag{1}
\mathbb{E}_\omega\left(\int_{t_0}^t(G_{\omega}(t'))^2\mathrm{d}t'\right)<+...
- 41
4
votes
0
answers
76
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Convergence in probability and differentiability of characteristic function
I am trying to prove the following:
If $(X_1+...+X_n)/n \xrightarrow{n \to \infty} m < \infty$ in probability then the characteristic function $\phi$ is differentiable in zero and $\phi'(0)=im$, ...
- 191
0
votes
1
answer
16
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Differentiability in zero of characteristic function and expected value of i.i.i random variables
I would like to prove the following implication:
If the characteristic function $\phi$ is differentiable in zero and $X_1\geq0$ a.s., then $E(X_1)=i\phi'(0)<\infty$.
(This is Exercise 15.4.4 iii) ...
- 191