VI. 2. 5
219
Remark. If (2.1.5) is replaced by
1-1
then L, converges in Pw-probability to 1, i.e. the distribution of log L, converges lo the degenerate normal distribution (0,0). This could be proved as follows: First, we note that Td given by (2.1.13) satisfies ET/ -* 0, and that considerations employed in the proofs of Lemmas 2.1.a and 2.1.b yield the relation E(Wd - Td)2 -» 0. Thus ElV/ -» 0, and, consequently, the distribution of Wd converges to the degenerate normal distribution (0,0). Thus it remains to show that LeCam's second lemma extends to this case, too. However, the degenerate convergence of the distribution of Wd entails that (1.3.10), (1.3.11) and (1.3.12) hold with a2 = 0 (LoĚVE (1955), p. 317). The rest of the proof needs no change, and wt obtain that (log Lv — W,) -* 0 in probability, and hence log Lt -* 0 in probability. Consequently the Theorem remains valid even for b = 0 and /(/„) < co, and every statistic has the same limiting distribution, if any, under qd as under pd. Furthermore, the theorem remains valid even if we replace (2.1.5) by/(/0) < co and
(15)                                  m-2)2Ší>3<co.
For, if it were not valid, there would exist a sequence {tfv} satisfying (15) and such that the theorem would not be valid for any of its subsequences. However, since we can draw a subsequence {dj} <= {dv} such that
this would contradict the theorem if /<', > 0, and the above extension of the theorem, if b\ = 0.
Tlie results or the present section and related results were adapted and generalized from HAjek (1962).
2.5.    Rank statistics for //,. Consider
where/„ is a known density symmetric about zero, and A satisfies
(2)                                       NA2 -ir,   0 < b2 < co.