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The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0, infty), is ergodic and irreversible. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. The aim of the present paper is to provide quantitative estimates for the exponential convergence to equilibrium, in terms of the total variation and Wasserstein distances.
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The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in $[0,infty)$, is ergodic and irreversible. It belongs to the Additive Increase Multiplicative Decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for the process and also for its embedded chain.
We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1leq ileq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $nlog n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $left(n-frac{1}{2}right)log n$.
In this paper, we investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We show that the mixing time is of order $left(n-frac{3}{2}right)log n$ and prove that there is a total variation cutoff for this shuffle.
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
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