You can find that information in the system log /var/log/syslog
and /var/log/messages
.
Ex. grep -i 'kill' /var/log/messages
.
Try to examine /var/log/*. You if don't get information anywhere there, then I think you are out of luck.
As triplee pointed out, your kill process will not be logged by default. But as you said, you don't have sudo access, I assume you must be working in some organisation. So there are chances that users activity are logged. So my answer might work.
Unfortunately, without a log, you cannot go back in time and retrieve a list of running processes.
Using a simple script it is possible to keep a running log of processes. With the log, you can go back and view what was running and what wasn't.
#!/bin/bash
mkdir -p "$HOME/ps_logs"
while true; do
ps aux > "$HOME/ps_logs/ps_$(date +%Y-%m-%d_%H:%M:%S).log"
sleep 60 # Logging interval in seconds.
done