# Scaling Relations in Gaussian Random Walks

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# Preliminary Discussion

For a time series *x*_{i} the deltas are assumed to be normally distributed, i.e.,

with mean zero and variance σ^{2}. Then for a sample consisting of *N* + 1 ticks

Also note that

assuming that *N* is divisible by *k* and there are *N* + 1 ticks. Introducing time stamps for every tick, i.e., *x*(*t*_{i}): = *x*_{i}, the deltas become a function of fixed time intervals Δ*t*:

Note that the number of observations *n* depends on the sample size and the implicit Δ*t*. E.g., if there are *N* observations for a specified , then for Δ*t* there are

observations and *k* = *N* / *n*. The variable can be set to one second without loss of generality because the series Δ*x*_{i} can always be interpreted as Δ*x*_{i}(Δ*t* = 1). Hence Eq. (3) becomes

Furthermore

It is apparent that the only source of variability in Eq. (7) is the number of observations .

If one takes the logarithm of *x*_{i}, the numerical result is

where is the median value of *x*_{i}. Note that the series of delta log (mid) prices, with , is distribute equivalently to a series of for constant spread, so the right-hand side of Eq. (8) is applicable, where *a*_{i} and *b*_{i} are the ask and bid price at time *t*_{i}, respectively.

It is interesting to note that compared to Eq. (2), Eq. (8) still has an explicit reference to the price series, although switching to log space eliminates all absolute price references.

# Scaling Laws - Part I

For the scaling law mentioned in U. Müller et. al, "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis", Journal of Banking & Finance, Elsevier, vol. 14(6), 1990, relating the average Δ*x*(Δ*t*), or the volatility, to the time interval of sampling Δ*t*,

the above mentioned calculations can be employed. Setting *p* = 2, Eqs. (9) and (7) yield

or

In terms of logarithmic deltas, Eq. (8) yields

# Scaling Law Relations - Part II

Recall from Eqs. (2), (6) and (8) that fixed time intervals are assumed, i.e., Δ*t* is constant. In analogy to Eq. (1), a new dimension of randomness can
be added to the process by letting the time intervals vary as well:

So this new time series, Δ*x*_{i}(Δ*t*_{i}), has two independent sources of variability: one for the price and another
for the time increments. Note however that for the calculations, like the left-hand side of Eq. (6), there is still
a fixed sampling period assumed, Δ*t*. This requires an interpolation scheme, if there is no price tick for the sampling
time *x*(*t*_{i} + Δ*t*). For the current analysis, previous-tick interpolation is employed.

It follows from Eq. (13) that the scaling properties will be impacted, as the number of observations in
Eq. (6) depends on Δ*t* and the scaling law of Eq. (9) now reads

yielding

For logarithmic returns, the corresponding result is