Limit theorems for nonnegative independent random variables with truncation

Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type

“…random variables. This also improves a limit theorem for St. Petersburg games given in [GK11] and [Nak15] for i.i.d. random variables with stronger conditions on the truncation sequence, see Theorem 2.9 and Remark 2.10.…”

“…setting with q restricted to 1/2. A combination of Theorem 1.2 and Corollary 1.1 of [Nak15] gives the limit result as in (18) but imposes a stronger condition on (f n ) than (17).…”

Limit Theorems for Super-heavy Tailed Random Variables with Truncation: Application to the Super-petersburg Game

“…(ii) Conditions of Eqs. (18) and (22) for O-super-heavy tailed distributions correspond to [24,Eqs. (13) and 23] for power laws with index α, respectively.…”

Some limit theorems for weighted negative quadrant dependent random variables with infinite mean

“…For any , let us define and Then from Lemma 2.2 it follows that and are both pairwise NQD, and Furthermore, let us define , then the limit (2.2) holds if we show and Using the proof of Lemma 2.2 in [4], we get and From Lemma 2.1 and Lemma 2.3, we have which implies (2.6). Similarly, for any , we have which yields (2.7).…”

“…The theorem states the condition of the mean’s existence is not necessary, and St. Petersburg game (see [2]) and Feller game (see [3]), which are well known as the typical examples, are formulated by a nonnegative random variable X with the tail probability for each fixed , where denotes Nakata [4] considered truncated random variables and studied strong laws of large numbers and central limit theorems in this situation. In [5], Nakata studied the weak laws of large numbers for weighted independent random variables with the tail probability (1.2) and explored the case that the decay order of the tail probability is −1.…”