What is Limit superior of events?

Example 21.24 (Translating “infinitely often” and “eventually”)

The complement of $\limsup_nA_n$ is, by de Morgan,

the event “eventually, $A_n$ fails” (written $\liminf_nA_n^c$). So “$A_n$ infinitely often” and “$A_n^c$ eventually” are complementary — keeping this dictionary straight prevents most quantifier accidents. Sample translations for coin tossing: “infinitely many heads” is $\limsup\{X_n = H\}$; “only finitely many runs of $100$ heads” is the complement of a limsup; “the running frequency converges to $\frac12$” is $\bigcap_j\bigcup_N\bigcap_{n\geq N}\{\abs{\widehat p_n - \tfrac12} < \tfrac1j\}$ — countable operations throughout, so all of these are honest events.

Example 21.26 (Infinite runs of heads)

Toss a fair coin forever, and let $A_n$ be the event “tosses $n, n+1, \dots, n + k - 1$ are all heads” (a run of $k$ heads starting at time $n$), for fixed $k$. The events $A_{jk}$ ($j = 1, 2, \dots$), depending on disjoint blocks of tosses, are independent, each of probability $2^{-k}$, and $\sum_j 2^{-k} = \infty$: by Borel–Cantelli 2, with probability $1$ infinitely many blocks are all-heads — every fixed pattern recurs infinitely often, almost surely. Conversely, if we let the run length grow, $B_n = {}$“a run of $2\log_2 n$ heads starts at $n$” has $\P(B_n) = n^{-2}$ summable, so almost surely only finitely many such long runs start: Borel–Cantelli calibrates precisely how long the longest runs are.