# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Weighted Poincaré inequality

Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...

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### Hashed coupon collector

The story:
A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection.
Every day, a random customer arrives and buys his favorite card (that is, each customer ...

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### Distribution of a linear pure-birth process's integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:
$$
Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k \bigg]
$$
where $(...

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### Proof of the existence of an optimal MDP with a stochastic reward signal?

I'm following Sutton's book on Reinforcement Learning, and he casually states that "There is always at least one policy that is better than
or equal to all other policies" for a given finite MDP. This ...

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**1**answer

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### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

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### Existence of stick breaking representations for random measures

The Dirichlet process has a roughly size ordered representation in terms of beta random variables, called a stick-breaking representation (Sethuraman, 1994). Similar results hold for the beta process, ...

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**1**answer

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### Continuity of green functions

I have a technical question on a continuity of green function.
Setting
Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$.
Let $X=(\{X_t\}_{t \ge 0},\{P_x\}...

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191 views

### Large deviation of random walk

1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk
$$
S_i = \sum_{j=1}^iX_j
$$
for $i=1,2,\ldots,n$.
I am looking for "good" exponential upper bounds ...

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**1**answer

63 views

### Filtration exercise

I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus".
Denote by $\mathcal{F}^X_{t_0}$ the natural filtration corresponding to a process $X:[0,\infty)\times ...

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### The distribution of the area of a region cut out by chordal SLE?

Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$.
For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...

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**1**answer

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### Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral
$$
\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t
$$
where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a ...

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### Defining weak solutions to infinitely many SDEs on the same probability space

Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...

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**1**answer

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### Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...

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### An application of Girsanov's Theorem

Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, $a$ has adapted derivative, ...

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### How to obtain mathematical expectation with the vector as random variable?

In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...