# Non Markovian Process Example Essay

Here are two examples, in discrete time and in continuous time, hopefully not too close from each other.

1. The process of the running maxima of a Markov process is in general not Markov.

1.1 Consider your favorite Markov process, say the standard symmetric random walk $(X_n)_{n\geqslant0}$ on the integer line, defined by $X_0=0$ and $X_n=Y_1+\cdots+Y_n$ for every $n\geqslant1$, where $(Y_n)_{n\geqslant1}$ is i.i.d. and symmetric Bernoulli, hence $\mathrm P(Y_n=1)=\mathrm P(Y_n=-1)=\frac12$. The process of the running maxima of $(X_n)_{n\geqslant0}$ is the process $(M_n)_{n\geqslant0}$ defined by $$M_n=\max\{X_k\,;\,0\leqslant k\leqslant n\}.$$ Then:

The process $(M_n)_{n\geqslant0}$ is not a Markov process, nor a Markov process of any higher order.

To see this, note that $M_{n+1}$ is either $M_n$ or $M_n+1$, and that the probability that $M_{n+1}=M_n+1$ depends on $$T_n=\max\{0\leqslant k\leqslant n\,;\,M_{n-k}=M_n\},$$ the time elapsed since the process $(X_n)_{n\geqslant0}$ last hit its current maximum $M_n$. Then, due to the symmetry of the increments of the random walk, $\mathrm P(M_{n+1}=M_n+1\,\mid\,\mathcal M_n)=u(T_n)$, where $\mathcal M_n=\sigma(M_k\,;\,0\leqslant k\leqslant n)$ and, for every $k\geqslant0$, $u(k)=\frac12\mathrm P(X_k=0)$. Thus, $u(2k-1)=0$ and $u(2k)=\frac12{2k\choose k}2^{-2k}$ for every $k\geqslant1$, hence the sequence $(u(2k))_{k\geqslant1}$ is decreasing. Since, for every $0\leqslant k\leqslant n$, $[T_n\geqslant k]=[M_{n-k}=M_n]$, this shows that the conditional probability that $[M_{n+1}=M_n+1]$ depends on the past in a possibly unlimited way.

1.2 The analogous continuous time process is the standard Brownian motion $(B_t)_{t\geqslant0}$. Consider $S_t=\sup\{B_s\,;\,0\leqslant s\leqslant t\}$. Then $(B_t)_{t\geqslant0}$ is a Markov process but $(S_t)_{t\geqslant0}$ is not.

2. Other examples without the Markov property are the processes of local times.

2.1 In the discrete setting, consider $$Z_n=\sum\limits_{k=1}^{n}[X_{2k}=0].$$ Then $Z_{n+1}$ is either $Z_n$ or $Z_n+1$, and the probability that $Z_{n+1}=Z_n+1$ depends on $$\max\{0\leqslant k\leqslant n\,;\,X_{2n-2k}=0\},$$ the time elapsed since the last zero of the random walk. For reasons similar to the ones explained for the maxima processes:

The process $(Z_n)_{n\geqslant0}$ is not a Markov process, nor a Markov process of any higher order.

2.2 The analogous process for the standard Brownian motion $(B_t)_{t\geqslant0}$ is the so-called local time at zero $(L_t)_{t\geqslant0}$. Likewise, $(L_t)_{t\geqslant0}$ is not a Markov process.

- Никакая это не паранойя. Этот чертов компьютер бьется над чем-то уже восемнадцать часов!» Конечно же, все дело в вирусе. Чатрукьян это чувствовал. У него не было сомнений относительно того, что произошло: Стратмор совершил ошибку, обойдя фильтры, и теперь пытался скрыть этот факт глупой версией о диагностике.

Чатрукьян не был бы так раздражен, если бы «ТРАНСТЕКСТ» был его единственной заботой.