Intereting Posts

Axiomatization of $\mathbb{Z}$
Are compact subsets of $\mathbb{R}$ with the compact complement topology closed?
On the difference between consecutive primes
Initial value problem for 2nd order ODE $y''+ 4y = 8x$
Complex matrix that commutes with another complex matrix.
What groups can G/Z(G) be?
$G$ modulo $N$ is a cyclic group when $G$ is cyclic
How do you calculate this limit?
What is $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$?
Irreducible polynomials have distinct roots?
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Conditional expectation on Gaussian random variables
Why is the 2 norm “special”?
Is greatest common divisor of two numbers really their smallest linear combination?
Hausdorff Dimension Calculation

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$

Note: This issue is the second edition of the book “The First Course in Stochastic Processes” Samuel Karlin author. Could anyone help me to solve it? I have no idea to answer it! I thank you!

- Prove X is a martingale
- fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?
- Bounded (from below) continuous local martingale is a supermartingale
- Extension of Dynkin's formula, conclude that process is a martingale.
- Time scaling of Brownian motion
- show that the solution is a local martingale iff it has zero drift

- How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?
- Inequality with Expectations
- How gaussian mixture models work?
- Why does this expected value simplify as shown?
- Odds of guessing suit from a deck of cards, with perfect memory
- Probability that a random binary matrix is invertible?
- Expectation of the maximum of gaussian random variables
- Independence and Events.
- What is the problem in this solution to the Two Child Problem?
- Find $p(B)$ given $P(A)$, $P(A\cup B)$, and one more piece of information

Let $T=\inf\{n:X_n>L\}$. Define $\mathbb{E}_x[\cdot]=\mathbb{E}[\cdot|X_0=x]$, $\mathcal{F}_n=\sigma(X_0,\dots,X_n)$, and $\mathcal{F}_T=\{A:A\bigcap\{T=n\}\in\mathcal{F}_n\text{ for all $n$}\}$.

Then since

$$\{X_n>L\text{ for some $0\le n\le m$ }\}=\{\max_{0\le n\le m}X_n>L\}=\{T\le m\}$$

we have

$$P_1\{\max_{0\le n\le m}X_n>L,X_m=0\}=P_1\{T\le m,X_m=0\}$$

$$=\mathbb{E}_1[1\{T\le m\}1\{X_m=0\}]$$

$$=\mathbb{E}_1[1\{T\le m\}\mathbb{E}_1[1\{X_{(m-T)+T}=0\}|\mathcal{F}_T]]$$

$$=^1\mathbb{E}_1[1\{T\le m\}\mathbb{E}_{X_T}[1\{X_{m-T}=0\}]$$

$$=^2\mathbb{E}_1[1\{T\le m\}(P_1\{X_{m-T}=0\})^{X_T}]]$$

$$\le^3 \mathbb{E}_1[1\{T\le m\}(P_1\{X_m=0\})^{L+1}]]\le (P_1\{X_m=0\})^{L+1}$$

$^1$ – by the strong Markov property;

$^2$ – $P_x(X_k=0)=P_1(X_k=0)^x$;

$^3$ – $X_T\ge L+1$ and $P\{X_k=0\}$ is non-decreasing in $k$.

Consequently,

$$P_1\{\max_{0\le n\le m}X_n>L|X_m=0\}=\frac{P_1\{\max_{0\le n\le m}X_n>L,X_m=0\}}{P_1\{X_m=0\}}\le (P_1\{X_m=0\})^L$$

- If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.
- $2^{k} \mid {2k \choose 0} \cdot 3^{0} + {2k \choose 2} \cdot 3^{1} + \cdots + {2k \choose 2i} \cdot 3^{i} + \cdots + {2k \choose 2k} \cdot 3^{k}$
- How to prove this limit composition theorem?
- How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$?
- Find values for probability density function
- Does a polynomial that's bounded below have a global minimum?
- Big-O proof showing that t(n) is O(1)
- Show finite complement topology is, in fact, a topology
- What is wrong in this proof: That $\mathbb{R}$ has measure zero
- A variant of the Knight's tour problem
- Primitive polynomials
- Effect of row operations on determinant for matrices in row form
- The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule
- If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that…)
- Ideal Coin Value Choices