Stochastic Time Synchronization Models Based on Agreement Algorithms

Abstract

We propose deterministic and stochastic models of clock synchronization in nodes of large distributed network locally coupled with a reliable external exact time server.

Appendix

1.1 Different Types of Convergence

Let us recall some facts about convergence of random variables and vectors. The sequence of random vectors $\varvec{\xi }_{n}$ with values in is said to be convergent in distribution (in law) to a random vector $\varvec{\xi }$ as $n\rightarrow \infty$ if

$$\begin{aligned} \mathsf {E}\,f(\varvec{\xi }_{n})\rightarrow \mathsf {E}\,f(\varvec{\xi }) \end{aligned}$$

(35)

for all bounded continuous functions . In such case we will write $\varvec{\xi }_{n}{\mathop {\rightarrow }\limits ^{d}}\varvec{\xi }$.

We will use notation $\left\langle \varvec{y},\varvec{z}\right\rangle :=\sum _{j=1}^{N}y_{j}z_{j}$ for the usual scalar product of two nonrandom vectors and $\Vert \varvec{y}-\varvec{z}\Vert$ for the Euclidian distance between $\varvec{y}$ and $\varvec{z}$. Let $L_{2}(d\mathsf {P})$ be a Euclidian space of random -valued vectors with the scalar product

$$\begin{aligned} \left\langle \varvec{\xi },\varvec{\eta }\right\rangle _{L_{2}(d\mathsf {P})}:=\mathsf {E}\,\left\langle \varvec{\xi },\varvec{\eta }\right\rangle =\mathsf {E}\,\sum _{j=1}^{N}\xi _{j}\eta _{j}. \end{aligned}$$

Thus the Euclidian norm in $L_{2}(d\mathsf {P})$ is given by $\left\| \varvec{\xi }\right\| _{L_{2}(d\mathsf {P})}^{2}=\sum _{j=1}^{N}\mathsf {E}\,\left| \xi _{j}\right| ^{2}$.

The sequence of random vectors $\varvec{\xi }_{n}$ with values in is said to be convergent to $\varvec{\xi }$ in $L_{2}(d\mathsf {P})$ (mean square convergent) if $\mathsf {E}\,\Vert \varvec{\xi }_{n}-\varvec{\xi }\Vert ^{2}\rightarrow 0$ as $n\rightarrow \infty$. Recall that independence of two zero mean vectors $\varvec{\xi }$ and $\varvec{\eta }$ implies their orthogonality: $\left\langle \varvec{\xi },\varvec{\eta }\right\rangle _{L_{2}(d\mathsf {P})}=0$.

It is well know that the mean square convergence is stronger than the convergence in distribution, i.e., $\varvec{\xi }_{n}{\mathop {\longrightarrow }\limits ^{L_{2}(d\mathsf {P})}}\varvec{\xi }\quad \Longrightarrow \quad \varvec{\xi }_{n}{\mathop {\rightarrow }\limits ^{d}}\varvec{\xi }$.

Lemma 4

Let $\varvec{\xi }_{n}$ and $\varvec{\eta }_{n}$ be two sequences of -valued random vectors. Assume that

• (i) $\left\| \varvec{\xi _{n}}-\varvec{\eta }_{n}\right\| _{L_{2}(d\mathsf {P})}\rightarrow 0$ as $n\rightarrow \infty$,

• (ii) $\varvec{\eta }_{n}{\mathop {\rightarrow }\limits ^{d}}\varvec{\eta }$ where is some random vector.

Then $\varvec{\xi _{n}}{\mathop {\rightarrow }\limits ^{d}}\varvec{\eta }$ as $n\rightarrow \infty$.

The proof of this lemma is rather straightforward. It is sufficient to check (35) only for bounded uniformly continuous functions. To do this one can use the Chebyshev inequality. We omit details.