Why is the auto created statistic on this column empty?
I was able to reproduce this, both with an empty statistic, and a populated statistic. I arranged for an automatic statistic to be created on an empty table, and the index was created later:

IF OBJECT_ID(N'dbo.Heap', N'U') IS NOT NULL
BEGIN
    DROP TABLE dbo.Heap;
END;
GO
CREATE TABLE dbo.Heap 
(
    id integer NOT NULL IDENTITY,
    val integer NOT NULL,
);
GO
-- Add 1000 rows
INSERT dbo.Heap
    WITH (TABLOCKX)
    (val)
SELECT
    SV.number
FROM master.dbo.spt_values AS SV
WHERE
    SV.[type] = N'P'
    AND SV.number BETWEEN 1 AND 1000;
GO
SELECT COUNT_BIG(*) 
FROM dbo.Heap AS H
JOIN dbo.Heap AS H2
    ON H2.id = H.id
WHERE H.id &gt; 0
AND H2.id &gt; 0;
GO
-- Empty table
TRUNCATE TABLE dbo.Heap;
GO
-- Repeat exact same query (RT = 500 + 0.2 * 1000 = 700)
GO
SELECT COUNT_BIG(*) 
FROM dbo.Heap AS H
JOIN dbo.Heap AS H2
    ON H2.id = H.id
WHERE H.id &gt; 0
AND H2.id &gt; 0;
GO
-- Add 1000 rows
INSERT dbo.Heap
    WITH (TABLOCKX)
    (val)
SELECT
    SV.number
FROM master.dbo.spt_values AS SV
WHERE
    SV.[type] = N'P'
    AND SV.number BETWEEN 1 AND 1000;
GO
-- Add index
ALTER TABLE dbo.Heap ADD 
    CONSTRAINT [PK dbo.Heap id]
    PRIMARY KEY NONCLUSTERED (id);
GO
SELECT
    S.[name],
    S.auto_created,
    DDSP.stats_id,
    DDSP.last_updated,
    DDSP.[rows],
    DDSP.rows_sampled,
    DDSP.steps,
    DDSP.unfiltered_rows,
    DDSP.modification_counter
FROM sys.stats AS S
CROSS APPLY sys.dm_db_stats_properties(S.[object_id], S.stats_id) AS DDSP
WHERE 
    S.[object_id] = OBJECT_ID(N'dbo.Heap', N'U');




I found that modifications continue to be tracked accurately on all non-empty duplicates, but only one statistic is updated automatically (regardless of the asynchronous setting).

Automatic statistics updates only occur when the query optimizer needs a particular statistic, and finds that it is out of date (an optimality-related recompile).

The optimizer chooses from duplicate statistics as mentioned in the Plan Caching and Recompilations in SQL Server 2012 paper:


  An issue not directly related to the topic of this document is: given multiple statistics on the same set of columns in the same order, how does the query optimizer decide which ones to load during query optimization? The answer is not simple, but the query optimizer uses such guidelines as: Give preference to recent statistics over older statistics; Give preference to statistics computed using FULLSCAN option to those computed using sampling; and so on.


The point being that the optimizer chooses one of the available duplicate statistics (the "best one), and that one is automatically updated if found to be stale.

I believe this is a change in behaviour from older releases - or at least the documentation suggests that all out-of-date statistics for an object would be updated as part of this process, but I have no idea when this changed. It was certainly after August 2013 when Matt Bowler posted Duplicate Statistics, which contains a handy AdventureWorks-based repo. That script now results in only one of the statistics objects being updated, while at the time both were.

The above explanation matches all the behaviours I observed while attempting to reproduce your scenario, but I doubt it is explicitly documented anywhere. It does seem like a sensible optimization, since there is little value in keeping duplicates fully updated.

This is probably all at a level of detail below that which Microsoft are willing to support. This also means it could change without notice.