Snapshot 1: too small a bandwidth for the kernel density estimate yields an estimate that varies too much

Snapshot 2: too large a bandwidth for the kernel density estimate yields an estimate that is too smooth

Snapshot 3: This plot shows

for

; see the definition of

below. As explained there, the functions

converge to the density of the random harmonic series. We see that, at

, the approximate densities are all very flat and near to

and, at

, the approximate densities are all near to

. As explained below, the true values of the density of the random harmonic series at

and

are slightly smaller than

and

, respectively. Below the plot, we show the values of the approximate densities at

; thus,

,

are all

, but

is the first one that is slightly smaller than

. We also see that there is a very small probability for

or

.

Recall that the usual harmonic series

diverges. Following Schmuland [3] (see also Morrison [1] and Nahin [2, pp. 23–24, 229–230]), let

,

be independent random variables;

is, for all

,

or

, each with probability

. Consider then the so-called random harmonic series

. It can be shown that

is a continuous random variable and the series converges almost surely. Further, although there is no theoretical upper (or lower) bound on

, we have

, so that the probability of a very large sum is exceedingly small; for example,

and

.

The density

of

is very flat near the origin, but actually the density does not have a flat top. Although the density is very near to

at

, actually the value is slightly smaller than

. Also, although the density is very near to

at

, actually the value is slightly smaller than

(the exact value is the so-called infinite cosine product integral divided by

). In the plot of the estimate of the density of

, we have shown a horizontal line at

and at

to check how well the estimated density has these theoretical properties

.

Schmuland defines

to be a uniform random variable with density

if

,

; these variables are independent. He shows that, almost surely,

. Let

be the density of the partial sum

. Then,

converges to the density

of

uniformly on

. The density

can be calculated either by convolutions or by inverting the characteristic function of the partial sum (which is the product of the characteristic functions of the corresponding

variables). Schmuland shows plots of

,

, and

. We have calculated

,

, …,

by inverting the characteristic function.

Schmuland shows that

for

, but

for

. Indeed, in the Demonstration we calculate that

. Schmuland also shows that

for

, but

for

.

[1] K. E. Morrison, "Cosine Products, Fourier Transforms, and Random Sums,"

*The American Mathematical Monthly*,

**102**(8), 1995, pp. 716–724.

[2] P. J. Nahin,

*Digital Dice: Computational Solutions to Practical Probability Problems*, Princeton, NJ: Princeton University Press, 2008.

[3] B. Schmuland, "Random Harmonic Series,"

*The American Mathematical Monthly*,

**110**(5), 2003, pp. 407–416.