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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

0 votes

Example of random walk in a random environment (RWRE) saying things on the environment

A couple of "one-dimensional" examples: https://arxiv.org/abs/1210.6328 and https://arxiv.org/abs/2209.00101
Serguei Popov's user avatar
2 votes
0 answers
95 views

Local martingale for a (two-dimensional) diffusion

Let X be a two-dimensional diffusion (a solution of dXt=f(Xt)dt+dBt, with B a standard two-dimensional Brownian motion) living on some open set ΛR2. Let $h:\Lambda …
3 votes

The Borel-Cantelli lemma for random walks

Just a quick remark about another counterexample that one may construct: take a Simple Random Walk (on the integer lattice) in dimension d3 (so it is transient); then, an infinite set can be re …
Serguei Popov's user avatar
5 votes

Random walk visiting a cylinder infinitely often

Well, for d2, the projection of Sn onto a hyperplane orthogonal to p is a zero-mean (d1)-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''ye …
Serguei Popov's user avatar
3 votes

Is the departure process of an infinite server queue independent of the arrival process?

The arrival and departure processes are obviously not independent: suppose that, with some very bad luck, no customers arrived to the system up to now; then (after the completion of the service of tho …
Serguei Popov's user avatar
1 vote
1 answer
143 views

The input and output processes in a single-server queue

Consider an M/M/1 queue with the arrival rate λ>0 and the service rate μ>λ (so that it is stable), in the stationary regime. Let At be the number of arrivals in the time interva …
0 votes
Accepted

The input and output processes in a single-server queue

Let ηt be the number of customers in the system at time t and ρ=λ/μ<1 be the load. It holds that η0+AtDt=ηt, so AtDt=ηtη0. Write $$ A_t D_t = \frac{ …
Serguei Popov's user avatar
2 votes
Accepted

CLT for Bernoulli RV with negative correlation

No, the CLT need not hold under these assumptions. Consider the following example: take p=1/2 for definiteness, and divide the (discrete) time into intervals I1=[1,2], In=(2n1,2n], $n\ge …
Serguei Popov's user avatar
3 votes
1 answer
202 views

Capacity of a unit disk with a small bump

Let Ar={zC:|z|1}{zC:|z1|r} be the unit disk with a small "bump" (I'm interested in the regime r0). What can be said about the logarithmic capaci …
7 votes
Accepted

Prove an anti-concentration inequality for a martingale

Basically, the proof goes along the following lines: (1) Take a small ε>0 and show that the expected exit time from the interval [εvl,εvl] is less tha …
Serguei Popov's user avatar
7 votes
Accepted

Spiral lattice random walk

It seems to me that this random walk is recurrent. Denote Yn=, where (X_n, n\geq 0) is your "spiral" walk. Then, as x\to \infty, my calculations imply that $$ \mathbb{E}(Y_{n+1}-Y_n\mid …
Serguei Popov's user avatar
12 votes
Accepted

The mean square distance of a random walk from the origin

Let us divide the (time) interval [0,n] into n/t subintervals of length t. Let us call the kth interval good, if, during that interval, the random walk spends time at least t/5 to the left o …
Serguei Popov's user avatar
5 votes
Accepted

Distribution of the area statistic for Catalan paths

Notice that the number of Catalan paths of area at least cn^{\frac{3}{2}+\varepsilon} is less than the number of all paths that deviate from the horizontal axis by at least $n^{\frac{1}{2}+\varepsil …
Serguei Popov's user avatar
1 vote
1 answer
108 views

Regularity of the entrance measure of SRW

Let S(n) be the discrete sphere of radius n (i.e., the internal boundary of the Euclidean discrete ball B(n)) centered in the origin, and consider a simple random walk starting at some $x\in\mat …
0 votes

A question in central limit theorem

Doesn't it follow from the Lévy's continuity theorem? I mean, consider the characteristic functions of S_{n-1}/s_n and X_n/s_n, the product of them converges to e^{-t^2/2}, so the characteristic …
Serguei Popov's user avatar

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