NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

Random Walk:

\[X_n = \sum_{i=1}^n X_i \]

with \[\Pr(X_i = 1) = p, \;\;\; \Pr(X_1 = -1) = q\]

where if \(p=q\) we have a simple, symmetric random walk, whereas if \(p \neq q\) we have a random walk with drift. See here.

Brownian motion is the continuous time equivalent.

Now, from here, a random walk is different from white noise, which is stationary, has constant variance, and no autocorrelation. The variance of a random walk is increasing.

In a random walk,

\[x_{t+1} = x_t + \varepsilon_{t+1}\]

with \(\varepsilon\) being white noise.

MULTIPLICATIVE STOCHASTIC PROCESS:

Will follow a lognormal or power-law distribution as explained here.

Geometric Brownian motion comes instead as the limit of a multiplicative random walk : look at an initial stock price, \(X_0,\) multiplied by factors \(\varepsilon_i\) at each time step:

\[X_n=X_0 \; \varepsilon_1\;\varepsilon_2 \;\cdots\; \varepsilon_n.\]

Therefore we will have

\[x_{t+1} = x_t \varepsilon_{t+1}\]

where \(\varepsilon\) can be a variable interest rate.

If we model these \(L_i\) as independent and identically distributed random variables, applying the logarithm allows us to write the equation additively:

\[\ln X_n=\ln X_0+ \sum_{i=1}^n\ln \varepsilon_i\]

The Central Limit Theorem applies, and \(\sum_{i=1}^n\ln \varepsilon_i\) is a sum of random variables, and it is distributed normally independently of the distribution of \(\varepsilon\). Consequently, \(\ln X_n\) is distributed normally as more terms are added, and \(\ln X_0\) becomes less significant. So \(X\) is log-normally distributed. Positive normal variables that multiply together are log-normal in distribution: investments, city size…

But, if there is a lower bound to \(X_n,\) then \(X\) will be distributed as a power law.

MARKOV PROCESS and AR(1):

From here a Markov process fulfills:

\[P(X_{t} = x_t | {\rm entire \ history \ of \ the \ process }) = P(X_{t}=x_t| X_{t-1}=x_{t-1})\]

and an AR(1) process is

\[X_{t} = c + \varphi X_{t-1} + \varepsilon_{t}\]

Hence, a Markov process is an AR(1) process.

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.