Ilya
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Registered User
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I am PhD student at TU Delft.
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Apr 4 |
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Time integral of a diffusion [Cross-posted](math.stackexchange.com/questions/350837/…) |
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Mar 29 |
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Nontrivial lower bounds on Cheeger inequalities for Markov chains The book by Meyn and Tweedie "Markov Chains and Stochastic Stability" discusses many criteria for the convergence of $\lim_n\|\mu K^n - \pi\|$. Since this difference can be regarded as a size of $K^n$ on functions orthogonal to $\pi$, I guess shall help you as well. |
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Mar 28 |
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stopping time expectation for gambler’s ruin @Douglas: completely agree, but since OP asked how to find a moment using MGT, I thought it's worth reminding how to do this. |
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Mar 20 |
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stopping time expectation for gambler’s ruin In case you know moment-generating function of $\tau$, you can simply use the fact that $$ \mathsf E[\tau^2] = m''(0) $$ where $m(t) = \mathsf E[\mathrm e^{\tau t}]$ is MGT of $\tau$ |
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Mar 15 |
awarded | ● Yearling |
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Mar 15 |
awarded | ● Yearling |
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Mar 15 |
revised |
Maximal probability of “infinitely often” over MDP added 358 characters in body |
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Mar 15 |
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Equivalence of two definitions of Lyapunov exponents Another website: math.stackexchange.com is perhaps more suitable for such questions |
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Mar 15 |
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The property of a Markov measure I would support HW in his question about the framework (even though it perhaps not related to probability) - seems to be similar to what I am doing. |
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Mar 15 |
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The property of a Markov measure By a Markov measure you mean, that for some stochastic kernel $K$ on the measurable space $(A,2^A)$ it holds that $P$ is the induced measure on $A^\Bbb N$? |
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Mar 15 |
asked | Maximal probability of “infinitely often” over MDP |
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Mar 15 |
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The property of a Markov measure What does an open set men in your context? |
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Mar 14 |
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How identify bounded Borel measurable functions Do you mean $\langle f,\mu\rangle := \int_S f(x)\mu(\mathrm dx)$ for any $\mu \in M(S)$ and bounded Borel $f$? |
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Mar 14 |
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Application of Stochastic Calculus I would suggest to take a look at the 1st chapter of "Stochastic Differential Equations: at introduction with applications" by B. Oksendal to start being awared of (a subset of) possible applications. |
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Mar 14 |
revised |
Sufficiency of stationary policy for negative stochastic dynamic programming added 71 characters in body; edited title |
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Mar 14 |
asked | Sufficiency of stationary policy for negative stochastic dynamic programming |
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Mar 5 |
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Expected distance of a random point to the convex hull of N other points Do you assume a fixed probability space (=dependence structure between $X$ and $Y$), or you only know distributions? |
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Feb 21 |
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Distribution of maximum of random walk conditioned to stay positive There is a way to solve this problem even for more general Markov processes, but could you please tell me more precise about your conditioning. In your case $\mathsf P(Y_k\geq 0\;\forall k) = 0$ so I guess you are interested rather in $$ \mathsf P(M_n \geq m|Y_k\geq 0 \; \forall 0\leq k\leq n) $$ am I right? |
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Feb 19 |
accepted | approximation methods in integral equations |
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Feb 18 |
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approximation methods in integral equations @rafiki: you are welcome |
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Feb 18 |
answered | approximation methods in integral equations |
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Feb 14 |
accepted | Upper bound concerning Snell envelope |
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Feb 14 |
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Upper bound concerning Snell envelope @Paul: I'm not sure whether it's true. Let $X\equiv \frac 12$ and let $p = 2$, then LHS is $\frac14$ and RHS is $\frac1{16}$, so your philosophy does not apply at least in such case. |
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Feb 14 |
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Upper bound concerning Snell envelope @Paul: are you sure you have to take $p$ power in the RHS two times? Or you are just asking the question in the math.stackexchange version? |
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Feb 14 |
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A terminal coalgebra of a certain functor on Mes Thanks, I'll take a look. |
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Feb 14 |
answered | Upper bound concerning Snell envelope |
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Feb 14 |
asked | A terminal coalgebra of a certain functor on Mes |
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Jan 21 |
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Coupling of vectors @Anthony: I agree - and I am pretty sure that one come up with a maximal coupling of $P$ and $\hat P$ doing it sequentially - I just wondered whether it's possible for the maximal coupling which satisfies one more additional assumption (aka $\gamma$-coupling according to Lindvall). |
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Jan 20 |
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Coupling of vectors As far as I've checked your example, it is correct - thanks again. I guess, I shall accept you answer - but could you suggest how to compute $\Bbb P(X_1 = \hat X_1)$ or maybe you know how to express $\Bbb P(X\in A,\hat X\in \hat A)$? |
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Jan 20 |
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Coupling of vectors Thanks a lot for this reference! |
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Jan 19 |
asked | Uniform transience criteria |
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Jan 19 |
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Coupling of vectors Thank you. Do you have a reference for the proof of the last formula? It's like a Fubini theorem, but I've never seen a proof of its version for kernels. Also, in your first example, do you mean that $P$ and $\hat P$ in place of $\mu$ and $\hat \mu$ - just to keep it consistent with the notation in OP? |
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Jan 19 |
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Coupling of vectors @Colin: thanks, fixed |
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Jan 19 |
revised |
Coupling of vectors edited body |
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Jan 19 |
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Tails of sums of Weibull random variables I guess, the following reference may be useful for you, it's also pretty recent: [An Introduction to Heavy-Tailed and Subexponential Distributions, 2011](springer.com/mathematics/probability/book/…). Perhaps, available in Russian as well. Furthermore, [this paper](math.nsc.ru/LBRT/v1/dima/publications/…) provides untight rates of convergence which hold however even in case you only have first two moments |
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Jan 18 |
revised |
Coupling of vectors added 403 characters in body |
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Jan 18 |
asked | Coupling of vectors |
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Jan 18 |
revised |
Approximating a hitting time for some state using the stationary distribution? deleted 2 characters in body |
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Jan 18 |
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Approximating a hitting time for some state using the stationary distribution? @Ori: thanks ${{}}$ |
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Jan 17 |
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a problem on DTMC It shall give you hint where to look at. You are dealing exactly with the concept of "taboo" probabilities, which are defined as $$ _rP^n(i,j):=\mathsf P_i(X_n = j,\tau_r>n) $$ where $\tau_r$ is the first hitting time of the state $r$. It is known that $_rP^n(i,j)$ is a sub-markovian kernel in case $r$ is accesible from some other states, but as you are doing conditioning, the kernels you are considering are indeed markovian kernels. I would suggest that you read about them in [Meyn and Tweedie](probability.ca/MT/BOOK.pdf). Just look for "taboo probabilit |
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Jan 17 |
answered | Approximating a hitting time for some state using the stationary distribution? |
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Jan 17 |
awarded | ● Organizer |
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Jan 17 |
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Convergence rate for product of stochastic matrices Number of iterations until $x(t)$ converges to a stable point - don't you mean that it shall reach such point in a finite number of steps? |
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Jan 17 |
revised |
Integral Fredholm equation of the second type added 25 characters in body |
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Jan 17 |
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Change of measure Markov process Took me quite some time to get it. Thanks (a bit too late, though) anyway! |
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Jan 11 |
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Topological conditions of Kolmogorov Extension Theorem Thanks a lot, Michael! Will you consider also putting this answer on a linked MSE question? |
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Jan 11 |
asked | Topological conditions of Kolmogorov Extension Theorem |
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Nov 30 |
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Is positive part of the kernel measurable? @Michael: thanks for the comment |
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Nov 30 |
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Symmetries of probability distributions @Colin: thank you very much, I shall check that |
