#### by Naresh Jain

Professor Kai Lai Chung’s contributions to probability theory have had a major influence on several areas of research in the subject. I will restrict my comments to some of his work in two areas, sums of independent random variables and the theory of Markov chains, which led to a significant amount of further work, including some of my own.

#### Sums of independent random variables

Kai Lai has made many outstanding contributions to this field, but
I would like to concentrate on his
1948
paper
[1].
If
__\( X_1,X_2,\dots \)__ is a sequence of real-valued independent random
variables, and
__\[ S_n = X_1+X_2+\dots +X_n
\quad\text{for }n\geq 1 \]__
denotes the
sequence of partial sums, then the almost-sure behavior of “large
values” of __\( \{S_n\} \)__ was very well understood. Indeed, in the
independent and identically distributed (i.i.d) case,
Hartman and Wintner
in 1941
[e2]
had already proved their celebrated law of
the iterated logarithm: __\( EX_1 = 0 \)__ and __\( EX^2_1 = 1 \)__
imply
__\begin{equation}
\limsup_{n\to\infty} \frac{S_n}{(2n\log\log n)^{1/2}} = 1 \ \text{ a.s.}
\label{eq1}\end{equation}__
In a more general non-i.d. context,
Feller
in 1943
[e3]
had written
almost the final word.
Chung
[1]
observed that, if __\( S_n \)__ is replaced
by __\( |S_n| \)__ in __\eqref{eq1}__, the assertion remains valid. However, if __\( S^*_n \)__
denotes __\( \max_{1\leq j\leq n} |S_j| \)__, then the behavior of “small
values” of __\( S^*_n \)__ had yet to be understood. He studied this
problem in
[1]
including the non-i.d. situation and proved that,
denoting __\( E(S^2_n) \)__ by __\( s^2_n \)__,
if __\( EX_j\equiv 0 \)__ then, under a
natural third moment assumption,
__\begin{equation}\liminf_{n\to\infty}
\frac{S^*_n}{s_n(\log\log s_n)^{-1/2}} = 8^{-1/2}\pi \ \text{ a.s.}
\label{eq2}\end{equation}__
To prove this result, he obtained the very profound probability
estimate
__\begin{equation}
\label{eq3}
P(S^*_n < cs_n)
={\frac{4}{\pi}}\sum^\infty_{j=0}\frac{(-1)^j}{2j+1}
\exp\Bigl(-{\frac{(2j+1)^2\pi^2}{8c^2}}\Bigr)
+ O\bigl((\log\log s_n/\log s_n)^{1/2}\bigr).
\end{equation}__
This
contains the probability distribution for a standard one-dimensional
Brownian motion process __\( \{B_t:t\geq 0\} \)__, if __\( S^*_n/s_n \)__ is replaced
by __\( \max_{0\leq t\leq 1} |B_t| \)__ on the left side, and the second term
is replaced by zero on the right side.

In the i.i.d. case, two questions arose after Chung’s work. The
first one was raised by Chung himself: If __\( EX_1 = 0 \)__ and __\( EX^2_1 = 1 \)__,
does __\eqref{eq2}__ hold without any further assumptions, with __\( s^2_n = n \)__?
The second natural question was to obtain the analogue of __\eqref{eq2}__ if
__\( X_1 \)__ is in the domain of attraction of a stable law.

As to the first question, several papers appeared on the subject
getting close to the conditions stipulated by Chung. The question
was finally settled in the affirmative by
Jain and Pruitt
in 1975
[e12].
The probability distribution given in __\eqref{eq3}__ by Chung played a
key role in the final solution. For the second question, if the
index of stability is __\( \alpha < 2 \)__, it was not even clear if one
should expect an analogue of __\eqref{eq2}__.

Fristedt had already observed that an
analogue
of __\eqref{eq1}__ could not
exist if __\( \alpha < 2 \)__. However,
Fristedt and Pruitt
[e10]
and Jain and Pruitt
[e11]
showed, under different conditions on __\( X_1 \)__, the
existence of a real sequence __\( \{b_n\} \)__ increasing to infinity such
that __\( \liminf (S^*_n/b_n) = c \)__ a.s., with __\( 0 < c < \infty \)__.
However, it
was not clear if the constant __\( c \)__ depended on the distribution of
__\( X_1 \)__ or on the limit distribution alone.

Donsker and Varadhan
[e13]
approached these problems through their
large-deviations probability estimates for stable processes, and
obtained explicit expressions for the limit constants.
Jain
[e14]
was then able to show, through an invariance principle, that the
limit constant for the __\( \liminf \)__ behavior of __\( (S^*_n/b_n) \)__ is the
same as for the relevant stable process obtained by
Donsker and Varadhan
[e13].

The story by no means ends here. For a two-parameter Brownian
motion __\( B(s,t) \)__ with __\( 0\leq s,t\leq 1 \)__, the leading term of the “small
ball” probability estimate for
__\[ P\bigl(\,\max_{0\leq s,t\leq 1}|B(s,t)|\leq c\bigr)
\quad\text{as }c\downarrow 0 \]__
is of great interest, and turned out
to be a challenging problem, solved by
Bass
[e16]
and
Talagrand
[e18].
Much work has been done by other authors for
parameter dimension larger than 2; these results, however, are not
as definitive as in the two-dimensional parameter case. Many
difficult questions still remain to be answered, and we can expect
these investigations to continue, all owing their origins to Chung’s
pioneering work.

#### Markov chains

It is difficult to imagine that anybody working in the area of
Markov processes would not be familiar with Chung’s monograph,
*Markov Chains with Stationary Transition Probabilities*
[3].
This
monograph deals with countable-state Markov chains in both
discrete time (Part I) and continuous time (Part II). Much of Kai
Lai’s fundamental work in the field is included in this monograph.
My comments will be confined to Part I. Here, for the first time,
Kai Lai gave a systematic exposition of the subject, which includes
classification of states, ratio ergodic theorems, and limit
theorems for functionals of the chain.

For a general state space, Doeblin had given a classification scheme
in a seminal paper
in 1937
[e1].
This and other work of Doeblin had
a major impact on the field, and led to further developments by
Chung
[2],
Doob
[e4],
Harris
[e5],
and Orey
[e6],
[e9].
In the early
1960s there were a number of basic ingredients of a general
state-space theory that would lead to an exact counterpart of Part I of
Chung
[3].
Much fundamental ground work, including positive
recurrence (the so-called Doeblin’s condition), was done by Doob
[e4].
Harris
[e5]
introduced his recurrence condition: There exists
a nonzero __\( \sigma \)__-finite measure __\( \varphi \)__ on the state space __\( S \)__
such that __\( \varphi (E) > 0 \)__ implies that starting from every __\( x\in S \)__,
__\( E \)__ is visited infinitely often a.s. He proved the existence of a
(unique) __\( \sigma \)__-finite invariant measure __\( \pi \)__ under this
condition. If __\( \pi (S) = +\infty \)__, one could call the process
null-recurrent, and one could ask if __\( \pi (E) < \infty \)__ implied that,
for every __\( x\in S \)__,
__\( P^n(x,E)\to 0 \)__ as __\( n\to\infty \)__; here, __\( P^n(x,E) \)__
denotes the __\( n \)__-step transition probability from __\( x \)__ to __\( E \)__. This
result was conjectured by Orey
[e6],
and was a natural extension to
the general state-space situation of the corresponding well-known
result for a countable state chain. Under Harris’s recurrence
condition, one could also ask if an analogous ratio ergodic theorem
was true; namely, if __\( \pi (F) > 0 \)__ and __\( \pi (E) < \infty \)__, does
__\begin{equation}
\label{eq4}
\sum^n_{j=1} P^j(x,E)\big/\sum^n_{j=1}P^j(y,F)
\,\to\, \pi (E)/\pi (F)
\quad\text{as }n\to\infty
\end{equation}__
for * all* __\( x,y\in S \)__? These questions were
answered in
[e7]
under Chung’s guidance. Chung gave an example,
reported in
[e7],
to show that in the general case (as opposed to
the countable state space case) __\eqref{eq4}__ is true only for __\( \pi \)__-almost
all __\( x,y \)__, and not for * all* __\( x,y \)__. The situation is different
when the state space is not countable, because one could stay in a
__\( \pi \)__-null set for a rather long time! A little later, Jain and
Jamison
[e8]
introduced an irreducibility condition: There exists a
nonzero __\( \sigma \)__-finite measure __\( \varphi \)__ on __\( S \)__ such that __\( \varphi
(E) > 0 \)__ implies that, starting from every __\( x\in S \)__, the process visits
__\( E \)__ with positive probability. In this paper they essentially
brought the program of
Doeblin
[e1]
and Chung
[2]
to completion.
Chung’s influence can be seen throughout these works. For other work
in the area one can refer to monographs by
Orey
[e9],
Revuz
[e15]
and
Meyn and Tweedie
[e17].