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      Questions tagged [probability-measures]

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      1answer
      99 views

      Finitely additive measure on Cartesian square of countable set

      Let $\mu$ be a probability measure on $(\omega, 2^\omega, \mu)$ measure space which is finitely additive and $\mu(A)=0$ for finite sets. We can define as usual $\mu^2$ on semiring $\mathcal{G}=\{A\...
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      1answer
      63 views

      Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

      Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
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      1answer
      99 views

      Measurable function

      Let $S$ be a countable set. Consider $X=S^{\mathbb{N}\cup\{0\}}$ the topological Markov shift equipped with the topology generated by the collection of cylinders. Denoted $\mathcal{B}$ as the Borel $\...
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      1answer
      75 views

      Variance of sum of $m$ dependent random variables

      I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here. Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
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      0answers
      25 views

      how to filter measurement noise out of a set of data [migrated]

      I need to find the best way to filter white noise from a set of data $(x_0,y_0),...,(x_n,y_n)$, $n>1000$, where $x:\text{ is a time variable}$ $y:\text{ is a physical quantity}$ The noise is ...
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      6answers
      967 views

      Maximizing the expectation of a polynomial function of iid random variables

      Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$. Question 1. What is ...
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      24 views

      Injectivity of covariance operator and full support of Gaussian measure

      Let $\gamma$ be a Gaussian measure on a separable Banach space $\mathcal X$ with the covariance operator $C\colon \mathcal X^* \to \mathcal X$ defined dy $$ \langle Cf, g\rangle = \int_{\mathcal X} \...
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      33 views

      Stochastic Control: Markovian restriction

      Consider a stochastic control problem, $$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$ where $X_t$ is a weak solution to the SDE $$dX_t = C_t dB_t, \quad X_0 = x \...
      1
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      1answer
      195 views

      Existence of a Lyapunov function for a log-concave measure

      Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
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      62 views

      How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?

      Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...
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      1answer
      155 views

      On the existence of a particular type of finite measure on $\mathbb N$

      Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
      3
      votes
      2answers
      192 views

      Example of measure for some algebra over N

      $\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
      3
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      1answer
      105 views

      Is there a coupling that induces a given coupling via a transition kernel?

      Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
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      0answers
      69 views

      Distribution of dot product of two unit complex random vectors [duplicate]

      Consider $u,v∈S^{M-1}\subset \mathbb{C}^M$ to be two independent unit norm random vectors on the $M?1$ dimensional complex sphere $S^{M?1}$. In addition, $u$ follows an isotropic distribution, i.e., $...
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      1answer
      77 views

      How does a statistical divergence change under a Lipschitz push-forward map?

      Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...

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