Standard deviation angle for random walk?

[Loren Booda]

We know that the standard deviation _[sig] for a random walk, represented by a net distance d, to be approximately the square root of the total number of steps N, each of length L, from the origin. I. e., d~N^1/2L~_[sig]L.

Does the angle attained after these steps also have a significant standard deviation?

 

[mathman]

no answer

All directions are equally probable. As a result, there is no meaningful mean or standard deviation.

 

[Loren Booda]

mathman, assume the Nth step lands at an angle _[the] from the origin. Can we say anything about the Kth (K>N) step's probable angle? (From what you said above, apparently, we can not.)

 

[mathman]

Which angle are you talking about - the total after k steps or the angle after k-n steps relative to where you were?

 

[Loren Booda]

I refer to the angle for the total [after] K steps from the origin, i. e., the absolute angle with the ray containing the Kth point, the origin as vertex, and a fixed, standard ray - together similar to the absolute angle considered after the initial N steps.

 

[mathman]

It is a tough highly non-linear problem. The cosine of the angle involves the ratio of two random variables. Then you need an arccos to get the angle itself.

You also have to be more precise as to the definition of the problem. How many dimensions for the random walk? Is it on a lattice or is any direction allowed for each step?

 

[Loren Booda]

mathman,

Please simplify the problem to two dimensions (hopefully generalizable to four dimensions). Let the lattice of allowed movement be cartesian or polar (hopefully generalizable to Riemannian). My goal is to incorporate this random walk as a unification between quantum (radiative) mechanics and general (manifold) relativity.

Thanks for your tenacity.

Loren

 

[mathman]

Defining the problem precisely is helpful, but it doesn't make it easy. Specifically, the cosine of the angle of the vector to the position after after k steps, relative to the vector to the position after n steps is the dot product of the k step vector with the n step vector divided by the product of the vectors magnitudes. Since the k step vector contains random variables, afffecting both the numerator and the denominator of the cosine, computing the variance of the cosine is tough. The variance of the angle is even tougher, since you have to take the arccos before computing the variance.

 

[Loren Booda]

My speculation for this problem involves the radial component of the object at K steps relative to that of the observer at N steps (representing the radiative evolution of quantum mechanics), and the angular component of the object at K steps relative to that of the observer at N steps (representing a spacetime manifold evolution of general relativity).

Any basis in physics?

 

[mathman]

(representing the radiative evolution of quantum mechanics), ... (representing a spacetime manifold evolution of general relativity).

I have no idea what you are talking about.

 

[Loren Booda]

Radiation travels randomly from point-to-point in vacuo according to quantum mechanics. To the observer it progresses as the evolving random walk radial (conventional) component. A plurality of particles manifests multifold random walk radial components.

The relative path of the random walk angular (unprecedented) components may represent for the observer the geodesics normal to a point source mass's gravitation. Such a point source radiates perpendicularly to its surrounding spacetime manifolds. An accumulation of masses manifests multifold random walk angular components.

 

[NateTG]

What do you want to define the θ to be if the walk returns to the origin? (This is likely on a lattice.)

In general, the angle is going to be unpredictable over a large number of steps.

For1 and 2 dimensional random walks, you can probably use the notion that the distance traveled in N steps is likely to be about sqrt(N) to make probabilist predictions about the rate of change of the angle with respect to the number of steps taken.

So if you have angle &theta; at N steps, and then you've got some K s.t. K < N, then you can probably make good predictions about &theta; at step N+K. As K increases relative to N, the predictive value decreases.

Strictly speaking K does not have to be smaller than N, but it makes for much weaker predictions.

 

[Hurkyl]

One of the things necessary to describe things like mean and standard deviation is that our random variable live in a domain like the real numbers.

Angle does not live in a domain like the real numbers. It has the cyclic property that 0 = 2&pi;. In particular, division is not a well-defined operation on angles... but division is an essential part of computing means and standard deviations.

As an example, what should the average of north and south be? We could say that north is &pi;/2 radians and south is 3&pi;/2 radians, and say that the average should be &pi; (west)... however if we said south is -&pi;/2 radians, then the average would be 0 (east).

Standard deviation is even less well-defined; not only does it rely on knowing what the mean is, it uses both division and square roots; square roots are very poorly defined for this domain!
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