Bernstein's concentration inequality

Formula: P(S_n >= n t) <= exp(-n t^2 / (2 sigma^2 + 2 M t / 3))

Usage

Bernstein(sigma2, M, n, t)

Arguments

sigma2

See Usage.

M

See Usage.

n

See Usage.

t

See Usage.

Value

List with bound, bound_two_sided, hoeffding, ratio, exponent.

References

Bernstein (1924). The original is in Russian; the inequality is stated in this form in every standard concentration-inequality reference.

Examples

Bernstein(sigma2 = 1, M = 1, n = 100, t = 0.1)
#> $bound
#> [1] 0.6163927
#> 
#> $bound_two_sided
#> [1] 1
#> 
#> $hoeffding
#> [1] 0.6065307
#> 
#> $ratio
#> [1] 1.01626
#> 
#> $exponent
#> [1] -0.483871
#> 
#> $method
#> [1] "Bernstein inequality"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"