Record Statistics for Discrete Random Walks

Question

I code a random walk of length 100 drawn from a LaplaceDistribution:

Accumulate[RandomVariate[LaplaceDistribution[0, 1], 10000]]

I am having trouble counting the number of record events in a discrete random walk. A record occurs at time t if the value of the random walk at time t is greater than all values of the walk for all times less than t. I want a function that will let count the number of records that occur in a walk of length n.

Thanks.

Answer 1

Here's one take:

path = Accumulate[RandomVariate[LaplaceDistribution[0, 1], 1000]];
records = FoldList[Max, path];

ListPlot[{
  path,
  records
  }]

records has lots of duplicates in it. It's a list where in each position we have the largest value up to that point. If we take the union of the values (or use DeleteDuplicates), we get the unique largest-so-far values, and if we count those we get the desired answer:

Length@Union[records]

52

Answer 2

Let's generate data from a random walk first

SeedRandom[42]
walkdata = Accumulate[RandomVariate[LaplaceDistribution[0, 1], 100]]

, then one way to get what you want is with a Fold:

Last@Fold[
  Function[{state,newvalue},
    With[{currentrecord=state[[1]],recordcounter=state[[2]]},
      If[newvalue > currentrecord,
        {newvalue,recordcounter+1},
        state
      ]
    ]
  ],
  {0,0},
  walkdata
]

33

During the fold we keep track of the currentrecord and the number of records (starting with {0,0}) and update it when we find a higher value, otherwise we keep the old. The endresult is the last record and the number of records we encountered from which we just keep the number of record updates (with Last).

Comparing it with C.E.s solution this mainly trades some code clarity (if that's most important definitely go with C.E.s version) for some potential speed up by saving the overhead of the Union function call. If you are dealing with long random walks or doing a lot of them this might become relevant. There is also the additional option to Compile if you need better performance.