Brownian motion has E W(t) = 0 and Cov(W(s), W(t)) = min(s, t), and
its sample paths are Holder-alpha for every alpha < 1/2. Those path
properties are exactly what bound its concentration function, so the
covariance identity is worth checking directly: the empirical
covariance of simulated random walks on a grid must reproduce
min(s, t) and, on the diagonal, the variance s.
Usage
Ghosalbmprior(n_grid = 200, n_sim = 400, seed = 42)
Arguments
- n_grid
Number of grid steps on [0, 1].
- n_sim
Number of simulated paths.
- seed
Seed for the deterministic draws.
Value
List with estimate (empirical covariance),
theory_min_st, cov_gap, var_gap,
method.
Details
Formula: W(t_i) = sum_{j <= i} Z_j / sqrt(n_grid), Z_j iid N(0, 1);
Cov(W(s), W(t)) = min(s, t).
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, Example 11.5.
Examples
# \donttest{
Ghosalbmprior()
#> $estimate
#> [1] 0.2766386
#>
#> $theory_min_st
#> [1] 0.25
#>
#> $cov_gap
#> [1] 0.02663859
#>
#> $var_gap
#> [1] 0.0216812
#>
#> $method
#> [1] "Brownian motion prior (GvdV 2017 Ex 11.5)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
# }
