# Ilya

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## Registered User

 Name Ilya Member for 2 years Seen May 17 at 14:28 Website Location Leiden Age 25
I am PhD student at TU Delft.
 Apr4 comment Time integral of a diffusion[Cross-posted](math.stackexchange.com/questions/350837/…) Mar29 comment Nontrivial lower bounds on Cheeger inequalities for Markov chainsThe 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. Mar28 comment 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. Mar20 comment stopping time expectation for gambler’s ruinIn 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 $\tau$ Mar15 awarded ● Yearling Mar15 awarded ● Yearling Mar15 revised Maximal probability of “infinitely often” over MDPadded 358 characters in body Mar15 comment Equivalence of two definitions of Lyapunov exponentsAnother website: math.stackexchange.com is perhaps more suitable for such questions Mar15 comment The property of a Markov measureI 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. Mar15 comment The property of a Markov measureBy 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 $A^\Bbb N$? Mar15 asked Maximal probability of “infinitely often” over MDP Mar15 comment The property of a Markov measureWhat does an open set men in your context? Mar14 comment How identify bounded Borel measurable functionsDo you mean $\langle f,\mu\rangle := \int_S f(x)\mu(\mathrm dx)$ for any $\mu \in M(S)$ and bounded Borel $f$? Mar14 comment 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. Mar14 revised Sufficiency of stationary policy for negative stochastic dynamic programmingadded 71 characters in body; edited title Mar14 asked Sufficiency of stationary policy for negative stochastic dynamic programming Mar5 comment Expected distance of a random point to the convex hull of N other pointsDo you assume a fixed probability space (=dependence structure between $X$ and $Y$), or you only know distributions? Feb21 comment Distribution of maximum of random walk conditioned to stay positiveThere 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? Feb19 accepted approximation methods in integral equations Feb18 comment approximation methods in integral equations@rafiki: you are welcome Feb18 answered approximation methods in integral equations Feb14 accepted Upper bound concerning Snell envelope Feb14 comment 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. Feb14 comment 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? Feb14 comment A terminal coalgebra of a certain functor on MesThanks, I'll take a look. Feb14 answered Upper bound concerning Snell envelope Feb14 asked A terminal coalgebra of a certain functor on Mes Jan21 comment 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). Jan20 comment Coupling of vectorsAs 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 $\Bbb P(X\in A,\hat X\in \hat A)$? Jan20 comment Coupling of vectorsThanks a lot for this reference! Jan19 asked Uniform transience criteria Jan19 comment Coupling of vectorsThank 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? Jan19 comment Coupling of vectors@Colin: thanks, fixed Jan19 revised Coupling of vectorsedited body Jan19 comment Tails of sums of Weibull random variablesI 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 Jan18 revised Coupling of vectorsadded 403 characters in body Jan18 asked Coupling of vectors Jan18 revised Approximating a hitting time for some state using the stationary distribution?deleted 2 characters in body Jan18 comment Approximating a hitting time for some state using the stationary distribution?@Ori: thanks ${{}}$ Jan17 comment a problem on DTMCIt 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 Jan17 answered Approximating a hitting time for some state using the stationary distribution? Jan17 awarded ● Organizer Jan17 comment Convergence rate for product of stochastic matricesNumber 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? Jan17 revised Integral Fredholm equation of the second typeadded 25 characters in body Jan17 comment Change of measure Markov processTook me quite some time to get it. Thanks (a bit too late, though) anyway! Jan11 comment Topological conditions of Kolmogorov Extension TheoremThanks a lot, Michael! Will you consider also putting this answer on a linked MSE question? Jan11 asked Topological conditions of Kolmogorov Extension Theorem Nov30 comment Is positive part of the kernel measurable?@Michael: thanks for the comment Nov30 comment Symmetries of probability distributions@Colin: thank you very much, I shall check that