regularity of zero point

We consider 1-d process X X(t)=bt+Jt+Mt where b is constant, M is a continuous martingale process with M(0)=0, and J is a symmestric α-stable process with its Levy symbol η(u)=−|u|α for some constant α∈[1,2). Its Levy measure is ν(y)=1|y|1+α. Consider τ=inf I want to know if the following claim is true. [Claim] \tau = 0 … Read more

Self-adjusting random walk

Let Xt be a random process such that X1=0Xt=Xt−1+{At,Xt−1≥0Bt,Xt−1<0 where At and Bt are i.i.d. random variables with the properties E[At]=−α,Var[At]=σ2E[Bt]=α,Var[Bt]=σ2. Question: I am looking for techniques to establish an upper bound on E[|Xt|] or E[X2t] which holds for all t∈N in terms of α and σ. I can write out E[X2t] as follows E[X2t]=E[(Xt−1+1Xt−1≥0At+1Xt−1<0Bt)2]=E[X2t−1]+E[(1Xt−1≥0At+1Xt−1<0Bt)2]+2E[Xt−1(1Xt−1≥0At+1Xt−1<0Bt)]=E[X2t−1]+α2+σ2−2E[|Xt−1|]α … Read more

Feynman-Kac formula and time-ordering for vector bundles

Let M be a compact Riemannian manifold and let dWyx;T(γ) denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from x to y in time T (in such a way that the integral over the function 1 gives back the heat kernel e−TΔ(x,y) on M). Now if f and g … Read more

concentration of functions of Gaussian processes

Let C∈Rn be a subset of the unit ball. Also let a1,a2,…,am∈Rn be i.i.d. random Gaussian vectors N(0,I) then by Gordon’s lemma we know supy∈C|1mm∑k=1(aTky)2−‖ holds with high probability as long as \begin{align*} m\ge c\frac{\omega^2(\mathcal{C})}{\delta^2}, \end{align*} for a fixed numerical constant c (in fact I think c is at most 9). Here \omega(\mathcal{C}) is the … Read more

Stochastic subgradient descent almost sure convergence

I was reading up on stochastic subgradient descent, and most sources i could find via google search give quick proofs on convergence in expectation and probability, and say that proofs of almost sure convergence exist. Can anyone point me to a proof of this claim? I would be especially interested in proving the following claim … Read more

Is there an equivalent line time-invariant system for a linear time-varying system with specific properties? [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 5 years ago. Improve this question Given a discrete-time linear time-varying system (LTV) x(k+1) = A(k) x(k) + B(k) u(k) where A(k) and B(k) are generated … Read more

Does a non-exchangeable empirical reverse-martingale exist?

Consider a possible finite sequence ξ1,ξ2,… of random variables and consider the measure-valued empirical process ηn=∑ni=1δξin,n=1,2,… Assume ηn is a reverse martingale, in the sense that (∫fdηn) is a reverse-martingale for every f≥0. Does it automatically hold that (ηn) is exchangeable? I.e., does it hold that for any f1,…,fn and permutation p of {1,…,n} E(∫f1η1⋯∫fnηn)=E(∫f1ηp(1)⋯∫fnηp(n))? … Read more

Construct Lyapunov-Foster function given invariant distribution

Consider a discrete time Markov chain on a countable state space which is irreducible, aperiodic, and has a given invariant distribution π. Then the chain is necessarily positive recurrent and ergodic (so the invariant distribution is unique and is the limiting distribution regardless of the initial distribution). Further assume that the chain is actually geometrically … Read more

What is the entropy of binomial decay?

Let’s play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this process, as a function of $N$, $T$, and $p$. The calculation is straightforward for $N=1$ and $T=\infty$. The probability $p_i$ that … Read more

Distribution of the inner product of Gaussian processes

Suppose that $X(s)$ and $Y(s)$ are second order real Gaussian processes (independent or not), respectively $X\sim \mathcal{N}(\mu_1,\mathcal{K}_1)$ and $Y\sim \mathcal{N}(\mu_2,\mathcal{K}_2)$, and $s\in S\subset\Bbb{R}^d$. Assume that $X$ and $Y$ are a.s. $L^2(S)$. Are there any standard results about the distribution of the inner product of $X$ and $Y$? I.e. define the real random variable $$ Z … Read more