# Generating a sequence perturbed by non-stationary white noise

I want to create a list of numbers that would come form a process in which each number depends on the past value and a white noise. But I'd like the white noise to be non-stationary in the mean; the mean should go up with every period.

Here's what I came up with, but it's only with a stationary white noise:

SeedRandom["série5"]
série5 =
Flatten @
NestList[
0.5*# + RandomVariate[NormalDistribution[2, 1], 1]&, RandomReal[], 10]


Any ideas?

• Can't FractionalBrownianMotionProcess and FractionalGaussianNoiseProcess be employed? Commented Sep 5, 2016 at 17:57
• @Karsten7. thanks Commented Sep 5, 2016 at 18:40

Different approaches using functional iteration, all resulting in the same output.

SeedRandom["série5"];
série5 = Block[{m = 2 - 0.1},
NestList[0.5*# + RandomVariate[NormalDistribution[m += 0.1, 1]] &, RandomReal[], 10]]


SeedRandom["série5"];
série5 =
NestList[{0.5*#[[1]] + RandomVariate[NormalDistribution[#[[2]], 1]], #[[2]] + 0.1} &,
{RandomReal[], 2}, 10][[All, 1]]


SeedRandom["série5"];
série5 =
FoldList[0.5*#1 + RandomVariate[NormalDistribution[#2, 1]] &, RandomReal[],
Range[2, 2.9, 0.1]]


SeedRandom["série5"];
série5 =
FoldList[0.5*#1 + #2 &, RandomReal[],
Array[RandomVariate[NormalDistribution[#, 1]] &, 10, {2, 2.9}]]


SeedRandom["série5"];
série5 =
SequenceFoldList[0.5*#1 + #2 &, {RandomReal[]},
Array[RandomVariate[NormalDistribution[#, 1]] &, 10, {2, 2.9}]]